diff --git "a/data/GeometryLeanBench.jsonl" "b/data/GeometryLeanBench.jsonl"
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+{"name": "IMO_1995_1", "uuid": "76e8da9a-7c63-40d2-9ce4-c1fa207614f6", "statement_id": "f59bf080-ea9c-4a2e-af9b-3c979a591411", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_1995_1 :\n  ∀ (A B C D X Y Z P M N O₁ O₂ : Point) (L LXY L_AM L_DN : Line) (C₁ C₂ : Circle),\n    distinctPointsOnLine A B L ∧ C.onLine L ∧ D.onLine L ∧ between A B C ∧ between B C D ∧\n    midpoint A O₁ C ∧ O₁.isCentre C₁ ∧ A.onCircle C₁ ∧ C.onCircle C₁ ∧\n    midpoint B O₂ D ∧ O₂.isCentre C₂ ∧ B.onCircle C₂ ∧ D.onCircle C₂ ∧\n    circlesIntersectsAtTwoPoints C₁ C₂ X Y ∧ distinctPointsOnLine X Y LXY ∧\n    Z.onLine LXY ∧ between B Z C ∧ P.onLine LXY ∧ P ≠ Z ∧\n    coll C P M ∧ M.onCircle C₁ ∧ M ≠ C ∧\n    coll B P N ∧ N.onCircle C₂ ∧ N ≠ B ∧\n    distinctPointsOnLine A M L_AM ∧ distinctPointsOnLine D N L_DN →\n    concurrent L_AM L_DN LXY := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_1995_1 :\n  ∀ (A B C D X Y Z P M N O₁ O₂ : Point) (L LXY L_AM L_DN : Line) (C₁ C₂ : Circle),\n    distinctPointsOnLine A B L ∧ C.onLine L ∧ D.onLine L ∧ between A B C ∧ between B C D ∧\n    midpoint A O₁ C ∧ O₁.isCentre C₁ ∧ A.onCircle C₁ ∧ C.onCircle C₁ ∧\n    midpoint B O₂ D ∧ O₂.isCentre C₂ ∧ B.onCircle C₂ ∧ D.onCircle C₂ ∧\n    circlesIntersectsAtTwoPoints C₁ C₂ X Y ∧ distinctPointsOnLine X Y LXY ∧\n    Z.onLine LXY ∧ between B Z C ∧ P.onLine LXY ∧ P ≠ Z ∧\n    coll C P M ∧ M.onCircle C₁ ∧ M ≠ C ∧\n    coll B P N ∧ N.onCircle C₂ ∧ N ≠ B ∧\n    distinctPointsOnLine A M L_AM ∧ distinctPointsOnLine D N L_DN →\n    concurrent L_AM L_DN LXY := by"}
+{"name": "Canada_1991_3", "uuid": "3b1a1023-ed42-4701-b405-1bae4d3c1d28", "statement_id": "2836913c-c516-425a-815c-b1be3515cf6a", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Canada_1991_3 :\n  ∀ (P O : Point) (Ω : Circle),\n    P.insideCircle Ω ∧ O.isCentre Ω →\n    ∃ (Γ : Circle), ∀ (A B M : Point),\n      (A.onCircle Ω ∧ B.onCircle Ω ∧ between A P B ∧ midpoint A M B) →\n      M.onCircle Γ := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Canada_1991_3 :\n  ∀ (P O : Point) (Ω : Circle),\n    P.insideCircle Ω ∧ O.isCentre Ω →\n    ∃ (Γ : Circle), ∀ (A B M : Point),\n      (A.onCircle Ω ∧ B.onCircle Ω ∧ between A P B ∧ midpoint A M B) →\n      M.onCircle Γ := by"}
+{"name": "Problem_2_30_USAMO_1997_2", "uuid": "58808507-e141-4634-b765-4976347d1ee6", "statement_id": "697ed648-2139-4a17-8f01-06d0b185569e", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Problem_2_30_USAMO_1997_2 :\n  ∀ (A B C D E F : Point)\n    (AB BC CA l1 l2 l3 EF FD DE lA lB lC : Line),\n    formTriangle A B C AB BC CA ∧\n    perpBisector B C l1 ∧ D.onLine l1 ∧\n    perpBisector C A l2 ∧ E.onLine l2 ∧\n    perpBisector A B l3 ∧ F.onLine l3 ∧\n    distinctPointsOnLine E F EF ∧\n    distinctPointsOnLine F D FD ∧\n    distinctPointsOnLine D E DE ∧\n    A.onLine lA ∧ perpLine lA EF ∧\n    B.onLine lB ∧ perpLine lB FD ∧\n    C.onLine lC ∧ perpLine lC DE →\n    concurrent lA lB lC := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Problem_2_30_USAMO_1997_2 :\n  ∀ (A B C D E F : Point)\n    (AB BC CA l1 l2 l3 EF FD DE lA lB lC : Line),\n    formTriangle A B C AB BC CA ∧\n    perpBisector B C l1 ∧ D.onLine l1 ∧\n    perpBisector C A l2 ∧ E.onLine l2 ∧\n    perpBisector A B l3 ∧ F.onLine l3 ∧\n    distinctPointsOnLine E F EF ∧\n    distinctPointsOnLine F D FD ∧\n    distinctPointsOnLine D E DE ∧\n    A.onLine lA ∧ perpLine lA EF ∧\n    B.onLine lB ∧ perpLine lB FD ∧\n    C.onLine lC ∧ perpLine lC DE →\n    concurrent lA lB lC := by"}
+{"name": "Euler's_Theorem", "uuid": "734221b1-9b59-4fc3-a4b6-f089ae384b2e", "statement_id": "66626ad3-ade8-4763-a00e-1fb47b8d125c", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Euler's_Theorem :\n  ∀ (A B C O I D : Point) (AB BC CA : Line),\n    formTriangle A B C AB BC CA ∧\n    circumCentre O A B C ∧\n    inCentre I A B C ∧\n    foot I D BC →\n    (|(O─I)| * |(O─I)| = |(O─A)| * (|(O─A)| - 2 * |(I─D)|)) ∧ (|(O─A)| ≥ 2 * |(I─D)|) := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Euler's_Theorem :\n  ∀ (A B C O I D : Point) (AB BC CA : Line),\n    formTriangle A B C AB BC CA ∧\n    circumCentre O A B C ∧\n    inCentre I A B C ∧\n    foot I D BC →\n    (|(O─I)| * |(O─I)| = |(O─A)| * (|(O─A)| - 2 * |(I─D)|)) ∧ (|(O─A)| ≥ 2 * |(I─D)|) := by"}
+{"name": "Lemma1_45", "uuid": "da543886-fadc-4467-bf28-cfc81d613408", "statement_id": "83df676c-6b89-4860-99b0-96b4f42307fb", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Lemma1_45 :\n  ∀ (A B C D E F M N I K : Point) (AB BC CA EF BI MN : Line) (Ω : Circle),\n    formTriangle A B C AB BC CA ∧\n    inCentre I A B C ∧\n    inCircle Ω AB BC CA ∧\n    tangentAtPoint D I BC Ω ∧\n    tangentAtPoint E I CA Ω ∧\n    tangentAtPoint F I AB Ω ∧\n    midpoint B M C ∧\n    midpoint A N C ∧\n    distinctPointsOnLine B I BI ∧\n    distinctPointsOnLine E F EF ∧\n    twoLinesIntersectAtPoint BI EF K ∧\n    distinctPointsOnLine M N MN →\n  perpPoint B K C K ∧\n  K.onLine MN := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Lemma1_45 :\n  ∀ (A B C D E F M N I K : Point) (AB BC CA EF BI MN : Line) (Ω : Circle),\n    formTriangle A B C AB BC CA ∧\n    inCentre I A B C ∧\n    inCircle Ω AB BC CA ∧\n    tangentAtPoint D I BC Ω ∧\n    tangentAtPoint E I CA Ω ∧\n    tangentAtPoint F I AB Ω ∧\n    midpoint B M C ∧\n    midpoint A N C ∧\n    distinctPointsOnLine B I BI ∧\n    distinctPointsOnLine E F EF ∧\n    twoLinesIntersectAtPoint BI EF K ∧\n    distinctPointsOnLine M N MN →\n  perpPoint B K C K ∧\n  K.onLine MN := by"}
+{"name": "Problem_3_28_Shortlist_2001_G1", "uuid": "19cac28a-74d3-473d-b36d-ebd93d66a12e", "statement_id": "49111f6e-8ea5-4792-a875-3c3382f6e176", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Problem_3_28_Shortlist_2001_G1 :\n  ∀ (A B C A1 B1 C1 P Q R S U V W X M N L K : Point)\n    (AB BC CA lPQ lQR lRS lSP lUV lVW lWX lXU lMN lNL lLK lKM lAA1 lBB1 lCC1 : Line),\n    formAcuteTriangle A B C AB BC CA ∧\n    square P Q R S lPQ lQR lRS lSP ∧ P.onLine BC ∧ Q.onLine BC ∧ R.onLine CA ∧ S.onLine AB ∧\n    midpoint P A1 R ∧ midpoint Q A1 S ∧\n    square U V W X lUV lVW lWX lXU ∧ U.onLine CA ∧ V.onLine CA ∧ W.onLine AB ∧ X.onLine BC ∧\n    midpoint U B1 W ∧ midpoint V B1 X ∧\n    square M N L K lMN lNL lLK lKM ∧ M.onLine AB ∧ N.onLine AB ∧ L.onLine BC ∧ K.onLine CA ∧\n    midpoint M C1 L ∧ midpoint N C1 K ∧\n    distinctPointsOnLine A A1 lAA1 ∧ distinctPointsOnLine B B1 lBB1 ∧ distinctPointsOnLine C C1 lCC1 →\n    concurrent lAA1 lBB1 lCC1 := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Problem_3_28_Shortlist_2001_G1 :\n  ∀ (A B C A1 B1 C1 P Q R S U V W X M N L K : Point)\n    (AB BC CA lPQ lQR lRS lSP lUV lVW lWX lXU lMN lNL lLK lKM lAA1 lBB1 lCC1 : Line),\n    formAcuteTriangle A B C AB BC CA ∧\n    square P Q R S lPQ lQR lRS lSP ∧ P.onLine BC ∧ Q.onLine BC ∧ R.onLine CA ∧ S.onLine AB ∧\n    midpoint P A1 R ∧ midpoint Q A1 S ∧\n    square U V W X lUV lVW lWX lXU ∧ U.onLine CA ∧ V.onLine CA ∧ W.onLine AB ∧ X.onLine BC ∧\n    midpoint U B1 W ∧ midpoint V B1 X ∧\n    square M N L K lMN lNL lLK lKM ∧ M.onLine AB ∧ N.onLine AB ∧ L.onLine BC ∧ K.onLine CA ∧\n    midpoint M C1 L ∧ midpoint N C1 K ∧\n    distinctPointsOnLine A A1 lAA1 ∧ distinctPointsOnLine B B1 lBB1 ∧ distinctPointsOnLine C C1 lCC1 →\n    concurrent lAA1 lBB1 lCC1 := by"}
+{"name": "Russian_Olympiad_2010", "uuid": "1b4c33d8-3d20-4afc-806e-c8f6a1a6c66b", "statement_id": "5c5b71a6-f964-47d8-8c4d-1e5e6ba7a3cd", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Russian_Olympiad_2010 :\n  ∀ (A B C X Y M : Point) (AB BC CA XY : Line),\n    formTriangle A B C AB BC CA ∧\n    |(A─B)| + |(B─C)| + |(C─A)| = 4 ∧\n    X.onLine AB ∧ Y.onLine AC ∧ |(A─X)| = 1 ∧ |(A─Y)| = 1 ∧\n    distinctPointsOnLine X Y XY ∧\n    M.onLine BC ∧ M.onLine XY\n    → (|(A─B)| + |(B─M)| + |(M─A)| = 2) ∨ (|(A─C)| + |(C─M)| + |(M─A)| = 2) := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Russian_Olympiad_2010 :\n  ∀ (A B C X Y M : Point) (AB BC CA XY : Line),\n    formTriangle A B C AB BC CA ∧\n    |(A─B)| + |(B─C)| + |(C─A)| = 4 ∧\n    X.onLine AB ∧ Y.onLine AC ∧ |(A─X)| = 1 ∧ |(A─Y)| = 1 ∧\n    distinctPointsOnLine X Y XY ∧\n    M.onLine BC ∧ M.onLine XY\n    → (|(A─B)| + |(B─M)| + |(M─A)| = 2) ∨ (|(A─C)| + |(C─M)| + |(M─A)| = 2) := by"}
+{"name": "Problem_4_42_USAMO_1988_4", "uuid": "324050e0-4409-4c35-9eb1-482dc64a5aa0", "statement_id": "987aa4c1-9058-4498-8ef6-f4687351b969", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Problem_4_42_USAMO_1988_4 :\n  ∀ (A B C I O O1 O2 O3 : Point) (AB BC CA : Line),\n    formTriangle A B C AB BC CA ∧\n    inCentre I A B C ∧\n    circumCentre O A B C ∧\n    circumCentre O1 I A B ∧\n    circumCentre O2 I B C ∧\n    circumCentre O3 I C A →\n    |(O─O1)| = |(O─O2)| ∧ |(O─O2)| = |(O─O3)| := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Problem_4_42_USAMO_1988_4 :\n  ∀ (A B C I O O1 O2 O3 : Point) (AB BC CA : Line),\n    formTriangle A B C AB BC CA ∧\n    inCentre I A B C ∧\n    circumCentre O A B C ∧\n    circumCentre O1 I A B ∧\n    circumCentre O2 I B C ∧\n    circumCentre O3 I C A →\n    |(O─O1)| = |(O─O2)| ∧ |(O─O2)| = |(O─O3)| := by"}
+{"name": "Problem_4_41_Hong_Kong_1998", "uuid": "c6139a46-e37f-44d4-9427-7f901ae316fe", "statement_id": "f3ef5704-9436-4d5a-ae35-fca714b7af78", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Problem_4_41_Hong_Kong_1998 :\n  ∀ (P Q R S H K : Point) (PQ QR RS SP HK QS : Line) (Ω :Circle),\n    cyclicQuadrilateral P Q R S PQ QR RS SP Ω ∧\n    ∠ P:S:R = ∟ ∧\n    distinctPointsOnLine P R PR ∧\n    foot Q H PR ∧\n    foot Q K SP ∧\n    distinctPointsOnLine H K HK ∧\n    distinctPointsOnLine Q S QS →\n  ∃ (X : Point), X.onLine HK ∧ X.onLine QS ∧ midpoint Q X S := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Problem_4_41_Hong_Kong_1998 :\n  ∀ (P Q R S H K : Point) (PQ QR RS SP HK QS : Line) (Ω :Circle),\n    cyclicQuadrilateral P Q R S PQ QR RS SP Ω ∧\n    ∠ P:S:R = ∟ ∧\n    distinctPointsOnLine P R PR ∧\n    foot Q H PR ∧\n    foot Q K SP ∧\n    distinctPointsOnLine H K HK ∧\n    distinctPointsOnLine Q S QS →\n  ∃ (X : Point), X.onLine HK ∧ X.onLine QS ∧ midpoint Q X S := by"}
+{"name": "IMO1985_1", "uuid": "e78aaae5-118e-41af-9c65-584525d1b0dc", "statement_id": "e8ec8c5e-de11-462b-bf7f-9a1ca58b6fab", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO1985_1 :\n  ∀ (A B C D K : Point) (Ω ω : Circle) (AB BC CD DA : Line),\n    cyclicQuadrilateral A B C D AB BC CD DA Ω ∧\n    K.onLine AB ∧\n    K.isCentre ω ∧\n    tangentLine BC ω ∧\n    tangentLine CD ω ∧\n    tangentLine DA ω →\n    |(A─D)| + |(B─C)| = |(A─B)| := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO1985_1 :\n  ∀ (A B C D K : Point) (Ω ω : Circle) (AB BC CD DA : Line),\n    cyclicQuadrilateral A B C D AB BC CD DA Ω ∧\n    K.onLine AB ∧\n    K.isCentre ω ∧\n    tangentLine BC ω ∧\n    tangentLine CD ω ∧\n    tangentLine DA ω →\n    |(A─D)| + |(B─C)| = |(A─B)| := by"}
+{"name": "Russian_Olympiad_1_41", "uuid": "832c9271-70be-42da-bf29-f487165e4602", "statement_id": "a4cb6af1-2dbb-4f86-b53e-f0354162054c", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Russian_Olympiad_1_41 :\n  ∀ (A B C D E F : Point) (AB BC CD DA : Line),\n    formQuadrilateral A B C D AB BC CD DA ∧\n    between B E F ∧\n    between B F C ∧\n    ∠ B:A:E = ∠ C:D:F ∧\n    ∠ E:A:F = ∠ F:D:E →\n    ∠ F:A:C = ∠ E:D:B := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Russian_Olympiad_1_41 :\n  ∀ (A B C D E F : Point) (AB BC CD DA : Line),\n    formQuadrilateral A B C D AB BC CD DA ∧\n    between B E F ∧\n    between B F C ∧\n    ∠ B:A:E = ∠ C:D:F ∧\n    ∠ E:A:F = ∠ F:D:E →\n    ∠ F:A:C = ∠ E:D:B := by"}
+{"name": "Problem_3_29_USA_TSTST_2011_4", "uuid": "b8466aa5-0cba-4613-89f1-f3c22d3a9222", "statement_id": "9a199651-933f-459f-ae39-f87481bc1992", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Problem_3_29_USA_TSTST_2011_4 :\n  ∀ (A B C D E F H O M N P Q R D E F : Point) (ω : Circle) (AB BC CA MN PQ : Line),\n    formAcuteTriangle A B C AB BC CA ∧\n    orthoCentre H A B C D E F AB BC CA ∧\n    circumCentre O A B C ∧\n    circumCircle ω A B C ∧\n    midpoint A M B ∧\n    midpoint A N C ∧\n    between M H P ∧\n    P.onCircle ω ∧\n    between N H Q ∧\n    Q.onCircle ω ∧\n    distinctPointsOnLine M N MN ∧\n    R.onLine MN ∧\n    R.onLine PQ →\n    ∠O:A:R = ∟ := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Problem_3_29_USA_TSTST_2011_4 :\n  ∀ (A B C D E F H O M N P Q R D E F : Point) (ω : Circle) (AB BC CA MN PQ : Line),\n    formAcuteTriangle A B C AB BC CA ∧\n    orthoCentre H A B C D E F AB BC CA ∧\n    circumCentre O A B C ∧\n    circumCircle ω A B C ∧\n    midpoint A M B ∧\n    midpoint A N C ∧\n    between M H P ∧\n    P.onCircle ω ∧\n    between N H Q ∧\n    Q.onCircle ω ∧\n    distinctPointsOnLine M N MN ∧\n    R.onLine MN ∧\n    R.onLine PQ →\n    ∠O:A:R = ∟ := by"}
+{"name": "JMO_2012_1", "uuid": "94c662df-d7c6-4252-ba4b-efb5eebbd927", "statement_id": "c272a042-de94-45a5-9711-f423571f4b0a", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem JMO_2012_1 :\n  ∀ (A B C P Q R S : Point) (AB BC CA : Line),\n    formTriangle A B C AB BC CA ∧\n    between A P B ∧ between A Q C ∧ |(A─P)| = |(A─Q)| ∧\n    between B S C ∧ between B R C ∧ between B S R ∧ S ≠ R ∧\n    ∠ B:P:S = ∠ P:R:S ∧ ∠ C:Q:R = ∠ Q:S:R →\n    cyclic P Q R S := by sorry\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem JMO_2012_1 :\n  ∀ (A B C P Q R S : Point) (AB BC CA : Line),\n    formTriangle A B C AB BC CA ∧\n    between A P B ∧ between A Q C ∧ |(A─P)| = |(A─Q)| ∧\n    between B S C ∧ between B R C ∧ between B S R ∧ S ≠ R ∧\n    ∠ B:P:S = ∠ P:R:S ∧ ∠ C:Q:R = ∠ Q:S:R →\n    cyclic P Q R S := by sorry"}
+{"name": "problem_CO20", "uuid": "3af5b371-ae8d-4006-bd05-8fc8c9241ea9", "statement_id": "c333fc28-2f87-40da-bb98-d49285032425", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem problem_CO20 :\n∀ (A B C I M P H Q : Point) (AB BC CA PH MH BH QH: Line),\n  formTriangle A B C AB BC CA ∧\n  |(A─B)| = |(B─C)| ∧\n  inCentre I A B C ∧\n  midpoint B M I ∧\n  between A P C  ∧\n  (|(A─P)| = 3 * |(P─C)|) ∧\n  between P I H ∧\n  distinctPointsOnLine P H PH ∧\n  distinctPointsOnLine M H MH ∧\n  distinctPointsOnLine B H BH ∧\n  distinctPointsOnLine Q H QH ∧\n  perpLine PH MH ∧\n  cyclic Q B C A ∧\n  |(A─Q)| = |(B─Q)|\n  → perpLine BH QH :=\nby\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem problem_CO20 :\n∀ (A B C I M P H Q : Point) (AB BC CA PH MH BH QH: Line),\n  formTriangle A B C AB BC CA ∧\n  |(A─B)| = |(B─C)| ∧\n  inCentre I A B C ∧\n  midpoint B M I ∧\n  between A P C  ∧\n  (|(A─P)| = 3 * |(P─C)|) ∧\n  between P I H ∧\n  distinctPointsOnLine P H PH ∧\n  distinctPointsOnLine M H MH ∧\n  distinctPointsOnLine B H BH ∧\n  distinctPointsOnLine Q H QH ∧\n  perpLine PH MH ∧\n  cyclic Q B C A ∧\n  |(A─Q)| = |(B─Q)|\n  → perpLine BH QH :=\nby"}
+{"name": "Simson_Line", "uuid": "ba4e357b-4102-4766-b75a-84cfbe0842c3", "statement_id": "88f017c0-ca67-42d1-9ff1-5aef2a7c0967", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Simson_Line :\n  ∀ (A B C P X Y Z : Point) (Ω : Circle) (AB BC CA : Line),\n    formTriangle A B C AB BC CA ∧\n    circumCircle Ω A B C ∧\n    P.onCircle Ω ∧\n    foot P X BC ∧\n    foot P Y CA ∧\n    foot P Z AB →\n    coll X Y Z := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Simson_Line :\n  ∀ (A B C P X Y Z : Point) (Ω : Circle) (AB BC CA : Line),\n    formTriangle A B C AB BC CA ∧\n    circumCircle Ω A B C ∧\n    P.onCircle Ω ∧\n    foot P X BC ∧\n    foot P Y CA ∧\n    foot P Z AB →\n    coll X Y Z := by"}
+{"name": "Canada1997_4", "uuid": "9fe613b6-d4c7-4364-91d4-7244e7a7973a", "statement_id": "14e488ae-8804-4a79-8022-ce8c87bdfb98", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Canada1997_4 :\n  ∀ (A B C D O : Point) (AB BC CD DA : Line),\n    parallelogram A B C D AB BC CD DA ∧\n    insideQuadrilateral O A B C D AB BC CD DA ∧\n    ∠ A:O:B + ∠ C:O:D = ∟ + ∟ →\n    ∠ O:B:C = ∠ O:D:C := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Canada1997_4 :\n  ∀ (A B C D O : Point) (AB BC CD DA : Line),\n    parallelogram A B C D AB BC CD DA ∧\n    insideQuadrilateral O A B C D AB BC CD DA ∧\n    ∠ A:O:B + ∠ C:O:D = ∟ + ∟ →\n    ∠ O:B:C = ∠ O:D:C := by"}
+{"name": "Lemma1_42", "uuid": "72a41a17-f764-44ec-b435-6a70611012a7", "statement_id": "909a29de-30fe-4977-a09b-744300157c7f", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Lemma1_42 :\n  ∀ (A B C I X Y Z D E F : Point) (AB BC CA XY YZ ZX : Line) (Ω : Circle),\n    formAcuteTriangle A B C AB BC CA ∧\n    circumCircle Ω A B C ∧\n    X.onCircle Ω ∧ |(X─B)| = |(X─C)| ∧ opposingSides X A BC ∧\n    Y.onCircle Ω ∧ |(Y─C)| = |(Y─A)| ∧ opposingSides Y B CA ∧\n    Z.onCircle Ω ∧ |(Z─A)| = |(Z─B)| ∧ opposingSides Z C AB ∧\n    inCentre I A B C ∧\n    foot X D YZ ∧\n    foot Y E ZX ∧\n    foot Z F XY ∧\n    coll X I D ∧\n    coll Y I E ∧\n    coll Z I F →\n  orthoCentre I X Y Z D E F YZ ZX XY := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Lemma1_42 :\n  ∀ (A B C I X Y Z D E F : Point) (AB BC CA XY YZ ZX : Line) (Ω : Circle),\n    formAcuteTriangle A B C AB BC CA ∧\n    circumCircle Ω A B C ∧\n    X.onCircle Ω ∧ |(X─B)| = |(X─C)| ∧ opposingSides X A BC ∧\n    Y.onCircle Ω ∧ |(Y─C)| = |(Y─A)| ∧ opposingSides Y B CA ∧\n    Z.onCircle Ω ∧ |(Z─A)| = |(Z─B)| ∧ opposingSides Z C AB ∧\n    inCentre I A B C ∧\n    foot X D YZ ∧\n    foot Y E ZX ∧\n    foot Z F XY ∧\n    coll X I D ∧\n    coll Y I E ∧\n    coll Z I F →\n  orthoCentre I X Y Z D E F YZ ZX XY := by"}
+{"name": "Iran_TST_2011_1", "uuid": "e6477945-e0ba-442a-a9d6-7994230ac208", "statement_id": "5f6d1818-1bcf-482e-b221-68f021b3944c", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Iran_TST_2011_1 :\n  ∀ (A B C M E F K L T : Point) (AB BC CA KL AT : Line),\n    formAcuteTriangle A B C AB BC CA ∧\n    ∠ A:B:C > ∠ B:C:A ∧\n    midpoint B M C ∧\n    foot B E CA ∧\n    foot C F AB ∧\n    midpoint M K E ∧\n    midpoint M L F ∧\n    distinctPointsOnLine K L KL ∧\n    T.onLine KL ∧\n    distinctPointsOnLine A T AT ∧\n    ¬(AT.intersectsLine BC) →\n    |(T─A)| = |(T─M)| := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Iran_TST_2011_1 :\n  ∀ (A B C M E F K L T : Point) (AB BC CA KL AT : Line),\n    formAcuteTriangle A B C AB BC CA ∧\n    ∠ A:B:C > ∠ B:C:A ∧\n    midpoint B M C ∧\n    foot B E CA ∧\n    foot C F AB ∧\n    midpoint M K E ∧\n    midpoint M L F ∧\n    distinctPointsOnLine K L KL ∧\n    T.onLine KL ∧\n    distinctPointsOnLine A T AT ∧\n    ¬(AT.intersectsLine BC) →\n    |(T─A)| = |(T─M)| := by"}
+{"name": "Numina_Geometry_1001", "uuid": "7c1b479d-49ff-4ebe-bbfd-d3e2ddead0cf", "statement_id": "5c28143c-9174-4bdb-86b0-08c27a005ff0", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Numina_Geometry_1001 :\n  ∀ (A B C D E M N : Point) (AB BC CA DE: Line),\n    formAcuteTriangle A B C AB BC CA ∧\n    foot A D BC  ∧\n    foot C E AB ∧\n    distinctPointsOnLine D E DE ∧\n    foot A M DE ∧\n    foot C N DE →\n    |(M─E)| = |(D─N)| := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Numina_Geometry_1001 :\n  ∀ (A B C D E M N : Point) (AB BC CA DE: Line),\n    formAcuteTriangle A B C AB BC CA ∧\n    foot A D BC  ∧\n    foot C E AB ∧\n    distinctPointsOnLine D E DE ∧\n    foot A M DE ∧\n    foot C N DE →\n    |(M─E)| = |(D─N)| := by"}
+{"name": "problem_MP19", "uuid": "2ca51235-bd06-454c-bd8b-78724f39051d", "statement_id": "c31a86bd-7459-42dd-80bf-00a439e3fd78", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem problem_MP19 :\n∀ (A B C D E F P : Point)\n  (AB BC CD DA BD AE CF AC : Line),\n  formQuadrilateral A B C D AB BC CD DA ∧\n  distinctPointsOnLine B D BD ∧\n  distinctPointsOnLine A E AE ∧\n  E.onLine BD ∧\n  perpLine AE BD ∧\n  distinctPointsOnLine C F CF ∧\n  twoLinesIntersectAtPoint CF BD F ∧\n  perpLine CF BD ∧\n  distinctPointsOnLine A C AC ∧\n  midpoint A P C\n→ |(P─E)| = |(P─F)| :=\nby\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem problem_MP19 :\n∀ (A B C D E F P : Point)\n  (AB BC CD DA BD AE CF AC : Line),\n  formQuadrilateral A B C D AB BC CD DA ∧\n  distinctPointsOnLine B D BD ∧\n  distinctPointsOnLine A E AE ∧\n  E.onLine BD ∧\n  perpLine AE BD ∧\n  distinctPointsOnLine C F CF ∧\n  twoLinesIntersectAtPoint CF BD F ∧\n  perpLine CF BD ∧\n  distinctPointsOnLine A C AC ∧\n  midpoint A P C\n→ |(P─E)| = |(P─F)| :=\nby"}
+{"name": "problem_HP97", "uuid": "9e0206fd-226c-4610-9860-d0a27a7e9263", "statement_id": "0d90477f-d1a9-4e96-9400-dff78d74c147", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem problem_HP97 :\n  ∀ (A B C O I : Point) (AB BC CA OI AI : Line),\n  formTriangle A B C AB BC CA ∧\n  circumCentre O A B C ∧\n  inCentre I A B C ∧\n  distinctPointsOnLine O I OI ∧\n  distinctPointsOnLine A I AI ∧\n  perpLine OI AI\n  → (|(A─B)| + |(A─C)|) = (|(B─C)| + |(B─C)|) := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem problem_HP97 :\n  ∀ (A B C O I : Point) (AB BC CA OI AI : Line),\n  formTriangle A B C AB BC CA ∧\n  circumCentre O A B C ∧\n  inCentre I A B C ∧\n  distinctPointsOnLine O I OI ∧\n  distinctPointsOnLine A I AI ∧\n  perpLine OI AI\n  → (|(A─B)| + |(A─C)|) = (|(B─C)| + |(B─C)|) := by"}
+{"name": "Numina_Geometry_925", "uuid": "0e819890-f618-44ee-8024-0f1c6198662d", "statement_id": "29c4ad3c-8f13-4352-83d2-d8a16e5ee489", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Numina_Geometry_925 :\n  ∀ (O M K N A B C D : Point) (Ω : Circle)\n    (OM MK KN NO AB CD : Line),\n    square O M K N OM MK KN NO ∧\n    O.isCentre Ω ∧\n    K.onCircle Ω ∧\n    threePointsOnLine A M B AB ∧\n    threePointsOnLine C N D CD ∧\n    A.onCircle Ω ∧\n    B.onCircle Ω ∧\n    C.onCircle Ω ∧\n    D.onCircle Ω →\n    |(A─M)| * |(M─B)| = |(C─N)| * |(N─D)| := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Numina_Geometry_925 :\n  ∀ (O M K N A B C D : Point) (Ω : Circle)\n    (OM MK KN NO AB CD : Line),\n    square O M K N OM MK KN NO ∧\n    O.isCentre Ω ∧\n    K.onCircle Ω ∧\n    threePointsOnLine A M B AB ∧\n    threePointsOnLine C N D CD ∧\n    A.onCircle Ω ∧\n    B.onCircle Ω ∧\n    C.onCircle Ω ∧\n    D.onCircle Ω →\n    |(A─M)| * |(M─B)| = |(C─N)| * |(N─D)| := by"}
+{"name": "Numina_Geometry_478", "uuid": "ecc3d7ff-e008-4243-b4e7-84bfef999397", "statement_id": "708a8863-5c99-482b-949f-a06001c04db4", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Numina_Geometry_478 : ∀ (A B C D P O : Point),\ntriangle P A B ∧\nbetween A P C ∧\nbetween B P D ∧\n|(P─A)| = |(P─D)| ���\n|(P─B)| = |(P─C)| ∧\ncircumCentre O P A B →\nperpPoint O P C D := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Numina_Geometry_478 : ∀ (A B C D P O : Point),\ntriangle P A B ∧\nbetween A P C ∧\nbetween B P D ∧\n|(P─A)| = |(P─D)| ∧\n|(P─B)| = |(P─C)| ∧\ncircumCentre O P A B →\nperpPoint O P C D := by"}
+{"name": "Numina_Geometry_617", "uuid": "4e6275c5-8aea-4cca-aeec-22e4e548aca8", "statement_id": "f87e3d53-e842-4332-82fd-06aeeb607b41", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Numina_Geometry_617 :\n  ∀ (a b c d e f : Point) (BD EF : Line),\n    triangle a b c ∧\n    coll a d c ∧\n    coll b e c ∧\n    coll d f c ∧\n    distinctPointsOnLine b d BD ∧\n    distinctPointsOnLine e f EF ∧\n    ∠ a:b:d = ∠ d:b:c ∧\n    ∠ b:d:e = ∠ e:d:c ∧\n    ∠ d:e:f = ∠ f:e:c ∧\n    ¬ (BD.intersectsLine EF)\n    → ∠ a:b:c = ∠ b:a:c + ∠ b:a:c := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Numina_Geometry_617 :\n  ∀ (a b c d e f : Point) (BD EF : Line),\n    triangle a b c ∧\n    coll a d c ∧\n    coll b e c ∧\n    coll d f c ∧\n    distinctPointsOnLine b d BD ∧\n    distinctPointsOnLine e f EF ∧\n    ∠ a:b:d = ∠ d:b:c ∧\n    ∠ b:d:e = ∠ e:d:c ∧\n    ∠ d:e:f = ∠ f:e:c ∧\n    ¬ (BD.intersectsLine EF)\n    → ∠ a:b:c = ∠ b:a:c + ∠ b:a:c := by"}
+{"name": "problem_HP71", "uuid": "7e0ecfe4-c081-43b0-8369-beea5d8750ac", "statement_id": "8fd1b199-8381-4493-949a-75ed3621373f", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem problem_HP71 :\n  ∀ (A B C D E F H O M N : Point)\n    (AB BC CA AD BE CF ED FD MN OH: Line),\n    formTriangle A B C AB BC CA ∧\n    distinctPointsOnLine A D AD ∧\n    distinctPointsOnLine B E BE ∧\n    distinctPointsOnLine C F CF ∧\n    twoLinesIntersectAtPoint BC AD D ∧ perpLine BC AD ∧\n    twoLinesIntersectAtPoint CA BE E ∧ perpLine CA BE ∧\n    twoLinesIntersectAtPoint AB CF F ∧ perpLine AB CF ∧\n    twoLinesIntersectAtPoint AD BE H ∧ H.onLine CF ∧\n    circumCentre O A B C ∧\n    orthoCentre H A B C ∧\n    distinctPointsOnLine D E ED ∧\n    distinctPointsOnLine F D FD ∧\n    twoLinesIntersectAtPoint ED AB M ∧\n    twoLinesIntersectAtPoint FD AC N ∧\n    distinctPointsOnLine O H OH ∧\n    distinctPointsOnLine M N MN\n  → perpLine OH MN\n:= by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem problem_HP71 :\n  ∀ (A B C D E F H O M N : Point)\n    (AB BC CA AD BE CF ED FD MN OH: Line),\n    formTriangle A B C AB BC CA ∧\n    distinctPointsOnLine A D AD ∧\n    distinctPointsOnLine B E BE ∧\n    distinctPointsOnLine C F CF ∧\n    twoLinesIntersectAtPoint BC AD D ∧ perpLine BC AD ∧\n    twoLinesIntersectAtPoint CA BE E ∧ perpLine CA BE ∧\n    twoLinesIntersectAtPoint AB CF F ∧ perpLine AB CF ∧\n    twoLinesIntersectAtPoint AD BE H ∧ H.onLine CF ∧\n    circumCentre O A B C ∧\n    orthoCentre H A B C ∧\n    distinctPointsOnLine D E ED ∧\n    distinctPointsOnLine F D FD ∧\n    twoLinesIntersectAtPoint ED AB M ∧\n    twoLinesIntersectAtPoint FD AC N ∧\n    distinctPointsOnLine O H OH ∧\n    distinctPointsOnLine M N MN\n  → perpLine OH MN\n:= by"}
+{"name": "Numina_Geometry_1339", "uuid": "bd5c13eb-a530-4e02-8320-78bb211ba1f5", "statement_id": "52621e91-1b03-4f51-a825-55b565796401", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Numina_Geometry_1339 :\n  ∀ (A B C D K L M : Point)\n    (AB BC CD DA BD AK : Line),\n    parallelogram A B C D AB BC CD DA ∧\n    distinctPointsOnLine B D BD ∧\n    distinctPointsOnLine A K AK ∧\n    between B K D ∧\n    twoLinesIntersectAtPoint AK CD L ∧\n    twoLinesIntersectAtPoint AK BC M\n  → |(A─K)| * |(A─K)| = |(K─L)| * |(K─M)| := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Numina_Geometry_1339 :\n  ∀ (A B C D K L M : Point)\n    (AB BC CD DA BD AK : Line),\n    parallelogram A B C D AB BC CD DA ∧\n    distinctPointsOnLine B D BD ∧\n    distinctPointsOnLine A K AK ∧\n    between B K D ∧\n    twoLinesIntersectAtPoint AK CD L ∧\n    twoLinesIntersectAtPoint AK BC M\n  → |(A─K)| * |(A─K)| = |(K─L)| * |(K─M)| := by"}
+{"name": "problem_HP97", "uuid": "ea2015f7-a93b-4d2f-9ba5-757e9b869c02", "statement_id": "fd30f612-9248-4bb0-9ee0-f58207c38ac0", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem problem_HP97 :\n  ∀ (A B C O I : Point) (AB BC CA OI AI : Line),\n  formTriangle A B C AB BC CA ∧\n  circumCentre O A B C ∧\n  inCentre I A B C ∧\n  distinctPointsOnLine O I OI ∧\n  distinctPointsOnLine A I AI ∧\n  perpLine OI AI\n  → (|(A─B)| + |(A─C)|) = (|(B─C)| + |(B─C)|) := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem problem_HP97 :\n  ∀ (A B C O I : Point) (AB BC CA OI AI : Line),\n  formTriangle A B C AB BC CA ∧\n  circumCentre O A B C ∧\n  inCentre I A B C ∧\n  distinctPointsOnLine O I OI ∧\n  distinctPointsOnLine A I AI ∧\n  perpLine OI AI\n  → (|(A─B)| + |(A─C)|) = (|(B─C)| + |(B─C)|) := by"}
+{"name": "Numina_Geometry_145", "uuid": "1f1bef85-7932-496f-9329-b431cd6aaf1d", "statement_id": "c5f72746-fd3f-4849-865f-1a3d0af07d8f", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Numina_Geometry_145 :\n  ∀ (A B C D E : Point) (O : Circle) (AD BC : Line),\n    (A.onCircle O) ∧\n    (B.onCircle O) ∧\n    (C.onCircle O) ∧\n    (D.onCircle O) ∧\n    (distinctPointsOnLine A D AD) ∧\n    (distinctPointsOnLine B C BC) ∧\n    (A ≠ B) ∧ (A ≠ C) ∧\n    (twoLinesIntersectAtPoint AD BC E) ∧\n    (|(A─B)| = |(A─C)|)\n    → |(A─E)| * |(A─D)| = |(A─C)| * |(A─C)| := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Numina_Geometry_145 :\n  ∀ (A B C D E : Point) (O : Circle) (AD BC : Line),\n    (A.onCircle O) ∧\n    (B.onCircle O) ∧\n    (C.onCircle O) ∧\n    (D.onCircle O) ∧\n    (distinctPointsOnLine A D AD) ∧\n    (distinctPointsOnLine B C BC) ∧\n    (A ≠ B) ∧ (A ≠ C) ∧\n    (twoLinesIntersectAtPoint AD BC E) ∧\n    (|(A─B)| = |(A─C)|)\n    → |(A─E)| * |(A─D)| = |(A─C)| * |(A─C)| := by"}
+{"name": "Parallel5", "uuid": "2fbf6961-2015-4886-b63f-1956c4400bd1", "statement_id": "7d2dfec2-333b-4690-8b80-258584dc6327", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Parallel5 : ∀ (S U V X R Y T W : Point) (SU VX RY : Line),\n  distinctPointsOnLine S U SU ∧\n  distinctPointsOnLine V X VX ∧\n  distinctPointsOnLine R Y RY ∧\n  twoLinesIntersectAtPoint SU RY T ∧ between S T U ∧ between T W Y ∧\n  twoLinesIntersectAtPoint VX RY W ∧ between V W X ∧ between R T W ∧\n  V.sameSide S RY ∧\n  X.sameSide U RY ∧\n  ∠ R:T:S + ∠ V:W:Y = ∟ + ∟ →\n  ¬ VX.intersectsLine SU :=\nby\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Parallel5 : ∀ (S U V X R Y T W : Point) (SU VX RY : Line),\n  distinctPointsOnLine S U SU ∧\n  distinctPointsOnLine V X VX ∧\n  distinctPointsOnLine R Y RY ∧\n  twoLinesIntersectAtPoint SU RY T ∧ between S T U ∧ between T W Y ∧\n  twoLinesIntersectAtPoint VX RY W ∧ between V W X ∧ between R T W ∧\n  V.sameSide S RY ∧\n  X.sameSide U RY ∧\n  ∠ R:T:S + ∠ V:W:Y = ∟ + ∟ →\n  ¬ VX.intersectsLine SU :=\nby"}
+{"name": "Quadrilateral4", "uuid": "f529de80-4715-46e8-aab0-f1ba439c8343", "statement_id": "c23fbaf0-7c28-491b-a588-01f50a9f4fc5", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Quadrilateral4 : ∀ (G H I J : Point) (GH IJ GJ GI : Line),\n  formQuadrilateral G H J I GH HJ IJ GI ∧ distinctPointsOnLine G J GJ ∧\n  |(I─J)| = |(G─H)| ∧\n  ¬ GH.intersectsLine IJ →\n  |(H─J)| = |(G─I)|  :=\nby\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Quadrilateral4 : ∀ (G H I J : Point) (GH IJ GJ GI : Line),\n  formQuadrilateral G H J I GH HJ IJ GI ∧ distinctPointsOnLine G J GJ ∧\n  |(I─J)| = |(G─H)| ∧\n  ¬ GH.intersectsLine IJ →\n  |(H─J)| = |(G─I)|  :=\nby"}
+{"name": "Congruent15", "uuid": "2125a3d9-e1fd-4dc5-9119-ea272ff45284", "statement_id": "5d15a75f-25e8-4b28-b80f-0412ab188249", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Congruent15 : ∀ (S T U V : Point) (ST TU UV SV SU : Line),\n  formTriangle S T U ST TU SU ∧\n  formTriangle S U V SU UV SV ∧\n  V.opposingSides T SU ∧\n  ∠ U:V:S = ∟ ∧\n  ∠ U:S:V = ∠ S:U:T ∧\n  ∠ S:T:U = ∟ →\n  |(U─V)| = |(S─T)| :=\nby\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Congruent15 : ∀ (S T U V : Point) (ST TU UV SV SU : Line),\n  formTriangle S T U ST TU SU ∧\n  formTriangle S U V SU UV SV ∧\n  V.opposingSides T SU ∧\n  ∠ U:V:S = ∟ ∧\n  ∠ U:S:V = ∠ S:U:T ∧\n  ∠ S:T:U = ∟ →\n  |(U─V)| = |(S─T)| :=\nby"}
+{"name": "Similar4", "uuid": "90f0b25d-cd70-4711-b52c-01651f927f8e", "statement_id": "8e6e4fef-5b46-4bec-b5f4-caee8a8f348d", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Similar4 : ∀ (E F G H I : Point) (EF GH EG FH : Line),\n  formTriangle E F I EF FH EG ∧\n  formTriangle G H I GH FH EG ∧\n  between E I G ∧\n  between H I F ∧\n  ∠ H:G:I = ∠ F:E:I →\n  similarTriangle E I F G I H :=\nby\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Similar4 : ∀ (E F G H I : Point) (EF GH EG FH : Line),\n  formTriangle E F I EF FH EG ∧\n  formTriangle G H I GH FH EG ∧\n  between E I G ∧\n  between H I F ∧\n  ∠ H:G:I = ∠ F:E:I →\n  similarTriangle E I F G I H :=\nby"}
+{"name": "Triangle18", "uuid": "4b122473-f481-4432-a554-bd0a6be7b2f2", "statement_id": "0e2f16ae-6bcc-4d71-9a6a-e446fb0862b9", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Triangle18 : ∀ (S T V R U W : Point) (ST TV VS RU UW WR : Line),\n  formTriangle S T V ST TV VS ∧\n  formTriangle R U W RU UW WR ∧\n  ∠ S:V:T = ∠ W:R:U ∧\n  ∠ V:S:T = ∠ R:W:U →\n  |(S─V)| / |(R─W)| = |(T─V)| / |(R─U)| :=\nby\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Triangle18 : ∀ (S T V R U W : Point) (ST TV VS RU UW WR : Line),\n  formTriangle S T V ST TV VS ∧\n  formTriangle R U W RU UW WR ∧\n  ∠ S:V:T = ∠ W:R:U ∧\n  ∠ V:S:T = ∠ R:W:U →\n  |(S─V)| / |(R─W)| = |(T─V)| / |(R─U)| :=\nby"}
+{"name": "similar_19", "uuid": "673d8a48-2796-4f19-933b-9cc3386860b8", "statement_id": "23a49c66-f6c0-4cf2-be23-03473c093fa6", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem similar_19 : ∀ (H I J K : Point) (HI IJ HJ IK : Line),\n  formTriangle H I K HI IK HJ ∧\n  formTriangle I J K IJ HJ IK ∧\n  between H K J ∧\n  ∠ H:K:I = ∟ ∧ ∠ I:K:J = ∟ ∧\n  ∠ H:I:J = ∟ →\n  similarTriangle I J K H I K :=\nby\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem similar_19 : ∀ (H I J K : Point) (HI IJ HJ IK : Line),\n  formTriangle H I K HI IK HJ ∧\n  formTriangle I J K IJ HJ IK ∧\n  between H K J ∧\n  ∠ H:K:I = ∟ ∧ ∠ I:K:J = ∟ ∧\n  ∠ H:I:J = ∟ →\n  similarTriangle I J K H I K :=\nby"}
+{"name": "Quadrilateral11", "uuid": "81a727ec-c5b9-4914-8099-c30b899cf1dd", "statement_id": "80993822-2a1c-4f6b-a7ff-489e3acb858c", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Quadrilateral11: ∀ (S T U V : Point) (ST UV SU TV: Line),\n  formQuadrilateral S T V U ST TV UV SU ∧\n  |(U─V)| = |(S─U)| ∧\n ∠ S:U:T = ∠ V:U:T  → |(S─T)| = |(T─V)|:=\nby\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Quadrilateral11: ∀ (S T U V : Point) (ST UV SU TV: Line),\n  formQuadrilateral S T V U ST TV UV SU ∧\n  |(U─V)| = |(S─U)| ∧\n ∠ S:U:T = ∠ V:U:T  → |(S─T)| = |(T─V)|:=\nby"}
+{"name": "Congruent8", "uuid": "0c2d9c34-945b-4355-b124-0e8207016b78", "statement_id": "ebfaa013-cf21-42de-8e4d-795ea2fd527f", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Congruent8 : ∀ (P Q R S T : Point) (PQ QR RS ST PT PS QS : Line),\n  formTriangle P S T PS ST PT ∧\n  formTriangle Q R S QR RS QS ∧\n  formTriangle P Q S PQ QS PS ∧\n  P.opposingSides R QS ∧ Q.sameSide R PS ∧\n  Q.opposingSides T PS ∧ P.sameSide T QS ∧\n  ∠ R:Q:S = ∠ S:P:T ∧\n  |(P─T)| = |(Q─R)| ∧\n  |(P─Q)| = |(Q─S)| ∧\n  |(Q─S)| = |(S─P)| →\n  congruentTriangle P S T Q S R:=\nby\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Congruent8 : ∀ (P Q R S T : Point) (PQ QR RS ST PT PS QS : Line),\n  formTriangle P S T PS ST PT ∧\n  formTriangle Q R S QR RS QS ∧\n  formTriangle P Q S PQ QS PS ∧\n  P.opposingSides R QS ∧ Q.sameSide R PS ∧\n  Q.opposingSides T PS ∧ P.sameSide T QS ∧\n  ∠ R:Q:S = ∠ S:P:T ∧\n  |(P─T)| = |(Q─R)| ∧\n  |(P─Q)| = |(Q─S)| ∧\n  |(Q─S)| = |(S─P)| →\n  congruentTriangle P S T Q S R:=\nby"}
+{"name": "Parallel16", "uuid": "cfac3934-2cd4-4ecc-bcb3-9774f62f66d0", "statement_id": "308509ab-a9b3-419e-9eb8-42b4fc676288", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Parallel16 : ∀ (T V W U X : Point) (TV TW UX VW : Line),\n  distinctPointsOnLine T V TV ∧\n  distinctPointsOnLine T W TW ∧\n  distinctPointsOnLine U X UX ∧\n  distinctPointsOnLine V W VW ∧\n  TV.intersectsLine UX ∧ U.onLine TV ∧ between T U V ∧\n  TW.intersectsLine UX ∧ X.onLine TW ∧ between T X W ∧\n  TV.intersectsLine VW ∧\n  TW.intersectsLine VW ∧\n  ∠ T:U:X = ∠ T:W:V ∧\n  ¬ UX.intersectsLine VW →\n  ∠ U:V:W = ∠ T:W:V :=\nby\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Parallel16 : ∀ (T V W U X : Point) (TV TW UX VW : Line),\n  distinctPointsOnLine T V TV ∧\n  distinctPointsOnLine T W TW ∧\n  distinctPointsOnLine U X UX ∧\n  distinctPointsOnLine V W VW ∧\n  TV.intersectsLine UX ∧ U.onLine TV ∧ between T U V ∧\n  TW.intersectsLine UX ∧ X.onLine TW ∧ between T X W ∧\n  TV.intersectsLine VW ∧\n  TW.intersectsLine VW ∧\n  ∠ T:U:X = ∠ T:W:V ∧\n  ¬ UX.intersectsLine VW →\n  ∠ U:V:W = ∠ T:W:V :=\nby"}
+{"name": "Triangle10", "uuid": "6b37b7b0-715d-455a-b085-f058478edafd", "statement_id": "7cd5be47-9687-4591-9f94-52c88668c459", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Triangle10 : ∀ (Q S T R U V : Point) (QS ST TQ RU UV VR : Line),\n  formTriangle Q S T QS ST TQ ∧\n  formTriangle R U V RU UV VR ∧\n  ∠ U:R:V = ∠ S:T:Q ∧\n  |(R─V)| / |(Q─T)| = |(R─U)| / |(S─T)| →\n  ∠ R:U:V = ∠ Q:S:T :=\nby\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Triangle10 : ∀ (Q S T R U V : Point) (QS ST TQ RU UV VR : Line),\n  formTriangle Q S T QS ST TQ ∧\n  formTriangle R U V RU UV VR ∧\n  ∠ U:R:V = ∠ S:T:Q ∧\n  |(R─V)| / |(Q─T)| = |(R─U)| / |(S─T)| →\n  ∠ R:U:V = ∠ Q:S:T :=\nby"}
+{"name": "IMO_2002_P2", "uuid": "ea7c3247-b416-413c-8eb8-f88209495c27", "statement_id": "6bb97734-9fdc-43a7-8f09-136274c9d4dd", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2002_P2 :\n  ∀ (A B C D E F I O : Point) (S : Circle) (DA AC L1 L2 : Line),\n    O.isCentre S ∧\n    B.onCircle S ∧ C.onCircle S ∧ A.onCircle S ∧\n    diameter B C O S ∧\n    ∠ A:O:B < 4 / 3 * ∟ ∧\n    D.onCircle S ∧\n    ∠ A:O:D = ∠ D:O:B ∧\n    distinctPointsOnLine D A DA ∧\n    distinctPointsOnLine A C AC ∧\n    O.onLine L1 ∧ ¬ L1.intersectsLine DA ∧ twoLinesIntersectAtPoint L1 AC I ∧\n    perpBisector O A L2 ∧\n    E.onLine L2 ∧ E.onCircle S ∧\n    F.onLine L2 ∧ F.onCircle S →\n  inCentre I C E F := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2002_P2 :\n  ∀ (A B C D E F I O : Point) (S : Circle) (DA AC L1 L2 : Line),\n    O.isCentre S ∧\n    B.onCircle S ∧ C.onCircle S ∧ A.onCircle S ∧\n    diameter B C O S ∧\n    ∠ A:O:B < 4 / 3 * ∟ ∧\n    D.onCircle S ∧\n    ∠ A:O:D = ∠ D:O:B ∧\n    distinctPointsOnLine D A DA ∧\n    distinctPointsOnLine A C AC ∧\n    O.onLine L1 ∧ ¬ L1.intersectsLine DA ∧ twoLinesIntersectAtPoint L1 AC I ∧\n    perpBisector O A L2 ∧\n    E.onLine L2 ∧ E.onCircle S ∧\n    F.onLine L2 ∧ F.onCircle S →\n  inCentre I C E F := by"}
+{"name": "IMO_2005_P5", "uuid": "73c0a67d-9c63-4d00-b49b-64a2c0a914e9", "statement_id": "35752313-a0ad-46d3-8949-f67a442b759d", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2005_P5 :\n  ∀ (A B C D E F P Q R X : Point)\n    (AB BC CD DA AC BD EF : Line)\n    (Ω1 Ω2 : Circle),\n    formQuadrilateral A B C D AB BC CD DA ∧\n    |(B─C)| = |(D─A)| ∧\n    BC.intersectsLine DA ∧\n    E.onLine BC ∧\n    F.onLine DA ∧\n    |(B─E)| = |(D─F)| ∧\n    twoLinesIntersectAtPoint AC BD P ∧\n    twoLinesIntersectAtPoint BD EF Q ∧\n    twoLinesIntersectAtPoint EF AC R ∧\n    circumCircle Ω1 P B C ∧\n    circumCircle Ω2 P D A ∧\n    circlesIntersectsAtTwoPoints Ω1 Ω2 P X →\n    cyclic P Q R X := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2005_P5 :\n  ∀ (A B C D E F P Q R X : Point)\n    (AB BC CD DA AC BD EF : Line)\n    (Ω1 Ω2 : Circle),\n    formQuadrilateral A B C D AB BC CD DA ∧\n    |(B─C)| = |(D─A)| ∧\n    BC.intersectsLine DA ∧\n    E.onLine BC ∧\n    F.onLine DA ∧\n    |(B─E)| = |(D─F)| ∧\n    twoLinesIntersectAtPoint AC BD P ∧\n    twoLinesIntersectAtPoint BD EF Q ∧\n    twoLinesIntersectAtPoint EF AC R ∧\n    circumCircle Ω1 P B C ∧\n    circumCircle Ω2 P D A ∧\n    circlesIntersectsAtTwoPoints Ω1 Ω2 P X →\n    cyclic P Q R X := by"}
+{"name": "IMO_2015_P3", "uuid": "e61b957b-2c5a-4d53-be9e-8352c94b5ac6", "statement_id": "d27ace8d-65ec-4669-8100-9a24b361a94b", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2015_P3 :\n  ∀ (A B C H F M K Q D E O₁ O₂ : Point) (AB BC CA l : Line) (Γ Δ Θ : Circle),\n    formAcuteTriangle A B C AB BC CA ∧\n    |(A─B)| > |(A─C)| ∧\n    circumCircle Γ A B C ∧\n    orthoCentre H A B C D E F AB BC CA ∧\n    midpoint B M C ∧\n    Q.onCircle Γ ∧ ∠ H:Q:A = ∟ ∧\n    K.onCircle Γ ∧ ∠ H:K:Q = ∟ ∧\n    A ≠ B ∧ B ≠ C ∧ C ≠ K ∧ K ≠ Q ∧ Q ≠ A →\n    circumCircle Δ K Q H ∧\n    circumCircle Θ F K M ∧\n    O₁.isCentre Δ ∧\n    O₂.isCentre Θ ∧\n    tangentAtPoint K O₁ l Δ ∧\n    tangentAtPoint K O₂ l Θ := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2015_P3 :\n  ∀ (A B C H F M K Q D E O₁ O₂ : Point) (AB BC CA l : Line) (Γ Δ Θ : Circle),\n    formAcuteTriangle A B C AB BC CA ∧\n    |(A─B)| > |(A─C)| ∧\n    circumCircle Γ A B C ∧\n    orthoCentre H A B C D E F AB BC CA ∧\n    midpoint B M C ∧\n    Q.onCircle Γ ∧ ∠ H:Q:A = ∟ ∧\n    K.onCircle Γ ∧ ∠ H:K:Q = ∟ ∧\n    A ≠ B ∧ B ≠ C ∧ C ≠ K ∧ K ≠ Q ∧ Q ≠ A →\n    circumCircle Δ K Q H ∧\n    circumCircle Θ F K M ∧\n    O₁.isCentre Δ ∧\n    O₂.isCentre Θ ∧\n    tangentAtPoint K O₁ l Δ ∧\n    tangentAtPoint K O₂ l Θ := by"}
+{"name": "IMO_2004_P1", "uuid": "58c0641b-8ed9-4039-8baa-4dc584fd34af", "statement_id": "38f85cb2-6412-49f6-b66f-eb3c9eb8144a", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2004_P1 :\n  ∀ (A B C M N O R X : Point) (AB BC CA : Line) (Ω : Circle),\n    formAcuteTriangle A B C AB BC CA ∧\n    |(A─B)| ≠ |(A─C)| ∧\n    diameter B C O Ω ∧\n    M.onCircle Ω ∧\n    N.onCircle Ω ∧\n    between A M B ∧\n    between A N C ∧\n    midpoint B O C ∧\n    ∠ B:A:R = ∠ R:A:C ∧\n    ∠ M:O:R = ∠ R:O:N →\n  ∃ (X : Point), X.onLine BC ∧ cyclic B M R X ∧ cyclic C N R X := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2004_P1 :\n  ∀ (A B C M N O R X : Point) (AB BC CA : Line) (Ω : Circle),\n    formAcuteTriangle A B C AB BC CA ∧\n    |(A─B)| ≠ |(A─C)| ∧\n    diameter B C O Ω ∧\n    M.onCircle Ω ∧\n    N.onCircle Ω ∧\n    between A M B ∧\n    between A N C ∧\n    midpoint B O C ∧\n    ∠ B:A:R = ∠ R:A:C ∧\n    ∠ M:O:R = ∠ R:O:N →\n  ∃ (X : Point), X.onLine BC ∧ cyclic B M R X ∧ cyclic C N R X := by"}
+{"name": "IMO_2006_P1", "uuid": "a80ea9e2-19cb-456f-b9b3-8495031ea8d7", "statement_id": "803c9e59-e22d-49b4-8d98-ec83b81b8e46", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2006_P1 :\n  ∀ (A B C I P : Point) (AB BC CA : Line),\n    formTriangle A B C AB BC CA ∧\n    inCentre I A B C ∧\n    insideTriangle P A B C AB BC CA ∧\n    ∠ P:B:A + ∠ P:C:A = ∠ P:B:C + ∠ P:C:B →\n    |(A─P)| ≥ |(A─I)| ∧ (|(A─P)| = |(A─I)| ↔ P = I) := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2006_P1 :\n  ∀ (A B C I P : Point) (AB BC CA : Line),\n    formTriangle A B C AB BC CA ∧\n    inCentre I A B C ∧\n    insideTriangle P A B C AB BC CA ∧\n    ∠ P:B:A + ∠ P:C:A = ∠ P:B:C + ∠ P:C:B →\n    |(A─P)| ≥ |(A─I)| ∧ (|(A─P)| = |(A─I)| ↔ P = I) := by"}
+{"name": "IMO_2018_P6", "uuid": "531b72aa-119d-4c81-bd7e-2d42b9fa7bc8", "statement_id": "29cf02a0-2aab-48b5-8c75-eab62324167e", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2018_P6 :\n  ∀ (A B C D X : Point) (AB BC CD DA : Line),\n    formQuadrilateral A B C D AB BC CD DA ∧\n    |(A─B)| * |(C─D)| = |(B─C)| * |(D─A)| ∧\n    insideQuadrilateral X A B C D AB BC CD DA ∧\n    ∠ X:A:B = ∠ X:C:D ∧\n    ∠ X:B:C = ∠ X:D:A →\n    ∠ B:X:A + ∠ D:X:C = ∟ + ∟ := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2018_P6 :\n  ∀ (A B C D X : Point) (AB BC CD DA : Line),\n    formQuadrilateral A B C D AB BC CD DA ∧\n    |(A─B)| * |(C─D)| = |(B─C)| * |(D─A)| ∧\n    insideQuadrilateral X A B C D AB BC CD DA ∧\n    ∠ X:A:B = ∠ X:C:D ∧\n    ∠ X:B:C = ∠ X:D:A →\n    ∠ B:X:A + ∠ D:X:C = ∟ + ∟ := by"}
+{"name": "IMO_2008_P6", "uuid": "a6d115d3-db4c-4985-b2fe-4d744ac53932", "statement_id": "b656ee35-5e0e-4b5e-95c3-f669403d1271", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2008_P6 :\n  ∀ (A B C D : Point) (AB BC CD DA CA : Line) (ω₁ ω₂ ω : Circle),\n    formQuadrilateral A B C D AB BC CD DA ∧ ¬ (|(A─B)| = |(B─C)|) ∧\n    inCircle ω₁ AB BC CA ∧ inCircle ω₂ CD DA CA ∧\n    tangentLine AB ω ∧ tangentLine BC ω ∧ tangentLine DA ω ∧ tangentLine CD ω →\n  ∃ (l₁ l₂ : Line) (X : Point),\n    tangentLine l₁ ω₁ ∧ tangentLine l₁ ω₂ ∧ tangentLine l₂ ω₁ ∧ tangentLine l₂ ω₂ ∧\n    twoLinesIntersectAtPoint l₁ l₂ X ∧ X.onCircle ω := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2008_P6 :\n  ∀ (A B C D : Point) (AB BC CD DA CA : Line) (ω₁ ω₂ ω : Circle),\n    formQuadrilateral A B C D AB BC CD DA ∧ ¬ (|(A─B)| = |(B─C)|) ∧\n    inCircle ω₁ AB BC CA ∧ inCircle ω₂ CD DA CA ∧\n    tangentLine AB ω ∧ tangentLine BC ω ∧ tangentLine DA ω ∧ tangentLine CD ω →\n  ∃ (l₁ l₂ : Line) (X : Point),\n    tangentLine l₁ ω₁ ∧ tangentLine l₁ ω₂ ∧ tangentLine l₂ ω₁ ∧ tangentLine l₂ ω₂ ∧\n    twoLinesIntersectAtPoint l₁ l₂ X ∧ X.onCircle ω := by"}
+{"name": "IMO_2016_P1", "uuid": "1055140e-83d9-4ea7-95e0-da6f625f935a", "statement_id": "11bc32f9-f1d5-45bc-a2f3-227132497b1a", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2016_P1 :\n  ∀ (B C F A D E M X : Point)\n    (BC CF FB AM MX XE EA BD FX ME : Line),\n    formTriangle B C F BC CF FB ∧\n    ∠ C:B:F = ∟ ∧\n    A.onLine CF ∧ between A F C ∧ |(F─A)| = |(F─B)| ∧\n    |(D─A)| = |(D─C)| ∧ ∠ D:A:C = ∠ C:A:B ∧\n    |(E─A)| = |(E─D)| ∧ ∠ E:A:D = ∠ D:A:C ∧\n    between C M F ∧ |(C─M)| = |(M─F)| ∧\n    parallelogram A M X E AM MX XE EA →\n    concurrent BD FX ME := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2016_P1 :\n  ∀ (B C F A D E M X : Point)\n    (BC CF FB AM MX XE EA BD FX ME : Line),\n    formTriangle B C F BC CF FB ∧\n    ∠ C:B:F = ∟ ∧\n    A.onLine CF ∧ between A F C ∧ |(F─A)| = |(F─B)| ∧\n    |(D─A)| = |(D─C)| ∧ ∠ D:A:C = ∠ C:A:B ∧\n    |(E─A)| = |(E─D)| ∧ ∠ E:A:D = ∠ D:A:C ∧\n    between C M F ∧ |(C─M)| = |(M─F)| ∧\n    parallelogram A M X E AM MX XE EA →\n    concurrent BD FX ME := by"}
+{"name": "IMO_2022_P4", "uuid": "a5b87002-a9a6-46bc-adc1-7561432803bd", "statement_id": "6249c790-26be-49a7-af3d-fae776e9d943", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2022_P4 :\n  ∀ (A B C D E T P Q R S : Point)\n    (AB BC CD DE EA CT DT : Line),\n    formPentagon A B C D E AB BC CD DE EA ∧\n    |(B─C)| = |(D─E)| ∧\n    T.sameSide A BC ∧ T.sameSide B CD ∧ T.sameSide C DE ∧ T.sameSide D EA ∧ T.sameSide E AB ∧\n    |(T─B)| = |(T─D)| ∧\n    |(T─C)| = |(T─E)| ∧\n    ∠ A:B:T = ∠ T:E:A ∧\n    distinctPointsOnLine C T CT ∧\n    distinctPointsOnLine D T DT ∧\n    P.onLine AB ∧ P.onLine CD ∧\n    Q.onLine AB ∧ Q.onLine CT ∧\n    between P B A ∧ between B A Q ∧\n    R.onLine EA ∧ R.onLine CD ∧\n    S.onLine EA ∧ S.onLine DT ∧\n    between R E A ∧ between E A S →\n    cyclic P S Q R := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2022_P4 :\n  ∀ (A B C D E T P Q R S : Point)\n    (AB BC CD DE EA CT DT : Line),\n    formPentagon A B C D E AB BC CD DE EA ∧\n    |(B─C)| = |(D─E)| ∧\n    T.sameSide A BC ∧ T.sameSide B CD ∧ T.sameSide C DE ∧ T.sameSide D EA ∧ T.sameSide E AB ∧\n    |(T─B)| = |(T─D)| ∧\n    |(T─C)| = |(T─E)| ∧\n    ∠ A:B:T = ∠ T:E:A ∧\n    distinctPointsOnLine C T CT ∧\n    distinctPointsOnLine D T DT ∧\n    P.onLine AB ∧ P.onLine CD ∧\n    Q.onLine AB ∧ Q.onLine CT ∧\n    between P B A ∧ between B A Q ∧\n    R.onLine EA ∧ R.onLine CD ∧\n    S.onLine EA ∧ S.onLine DT ∧\n    between R E A ∧ between E A S →\n    cyclic P S Q R := by"}
+{"name": "IMO_2001_P1", "uuid": "ee8a8a08-a6f8-4e5c-9258-5102d8dd767c", "statement_id": "3e9399ef-1e62-4ab2-95f9-7d1acf8f24dc", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2001_P1 :\n  ∀ (A B C P O : Point) (AB BC CA : Line),\n    formAcuteTriangle A B C AB BC CA ∧\n    foot A P BC ∧\n    circumCentre O A B C ∧\n    ∠ A:C:B ≥ ∠ C:B:A + ∟/3 →\n    ∠ B:A:C + ∠ C:O:P < ∟ := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2001_P1 :\n  ∀ (A B C P O : Point) (AB BC CA : Line),\n    formAcuteTriangle A B C AB BC CA ∧\n    foot A P BC ∧\n    circumCentre O A B C ∧\n    ∠ A:C:B ≥ ∠ C:B:A + ∟/3 →\n    ∠ B:A:C + ∠ C:O:P < ∟ := by"}
+{"name": "IMO_2023_P6", "uuid": "3af97993-d941-43a1-b577-787a43fba2aa", "statement_id": "9bf902e0-1901-40e2-80df-9db1aa3a01ea", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2023_P6 :\n  ∀ (A B C A1 B1 C1 A2 B2 C2 : Point)\n    (AB BC CA l1 l2 l3 l4 l5 l6 : Line),\n    formTriangle A B C AB BC CA ∧\n    |(A─B)| = |(B─C)| ∧ |(B─C)| = |(C─A)| ∧\n    insideTriangle A1 A B C AB BC CA ∧\n    insideTriangle B1 A B C AB BC CA ∧\n    insideTriangle C1 A B C AB BC CA ∧\n    |(B─A1)| = |(A1─C)| ∧\n    |(C─B1)| = |(B1─A)| ∧\n    |(A─C1)| = |(C1─B)| ∧\n    ∠ B:A1:C + ∠ C:B1:A + ∠ A:C1:B = (16/3) * ∟ ∧\n    distinctPointsOnLine B C1 l1 ∧ distinctPointsOnLine C B1 l2 ∧\n    A2.onLine l1 ∧ A2.onLine l2 ∧\n    distinctPointsOnLine C A1 l3 ∧ distinctPointsOnLine A C1 l4 ∧\n    B2.onLine l3 ∧ B2.onLine l4 ∧\n    distinctPointsOnLine A B1 l5 ∧ distinctPointsOnLine B A1 l6 ∧\n    C2.onLine l5 ∧ C2.onLine l6 ∧\n    |(A1─B1)| ≠ |(B1─C1)| ∧ |(B1─C1)| ≠ |(C1─A1)| ∧ |(C1─A1)| ≠ |(A1─B1)| →\n  ∃ (Ω1 Ω2 Ω3 : Circle) (X Y : Point),\n    circumCircle Ω1 A A1 A2 ∧ circumCircle Ω2 B B1 B2 ∧ circumCircle Ω3 C C1 C2 ∧\n    X.onCircle Ω1 ∧ X.onCircle Ω2 ∧ X.onCircle Ω3 ∧\n    Y.onCircle Ω1 ∧ Y.onCircle Ω2 ∧ Y.onCircle Ω3 ∧\n    X ≠ Y := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2023_P6 :\n  ∀ (A B C A1 B1 C1 A2 B2 C2 : Point)\n    (AB BC CA l1 l2 l3 l4 l5 l6 : Line),\n    formTriangle A B C AB BC CA ∧\n    |(A─B)| = |(B─C)| ∧ |(B─C)| = |(C─A)| ∧\n    insideTriangle A1 A B C AB BC CA ∧\n    insideTriangle B1 A B C AB BC CA ∧\n    insideTriangle C1 A B C AB BC CA ∧\n    |(B─A1)| = |(A1─C)| ∧\n    |(C─B1)| = |(B1─A)| ∧\n    |(A─C1)| = |(C1─B)| ∧\n    ∠ B:A1:C + ∠ C:B1:A + ∠ A:C1:B = (16/3) * ∟ ∧\n    distinctPointsOnLine B C1 l1 ∧ distinctPointsOnLine C B1 l2 ∧\n    A2.onLine l1 ∧ A2.onLine l2 ∧\n    distinctPointsOnLine C A1 l3 ∧ distinctPointsOnLine A C1 l4 ∧\n    B2.onLine l3 ∧ B2.onLine l4 ∧\n    distinctPointsOnLine A B1 l5 ∧ distinctPointsOnLine B A1 l6 ∧\n    C2.onLine l5 ∧ C2.onLine l6 ∧\n    |(A1─B1)| ≠ |(B1─C1)| ∧ |(B1─C1)| ≠ |(C1─A1)| ∧ |(C1─A1)| ≠ |(A1─B1)| →\n  ∃ (Ω1 Ω2 Ω3 : Circle) (X Y : Point),\n    circumCircle Ω1 A A1 A2 ∧ circumCircle Ω2 B B1 B2 ∧ circumCircle Ω3 C C1 C2 ∧\n    X.onCircle Ω1 ∧ X.onCircle Ω2 ∧ X.onCircle Ω3 ∧\n    Y.onCircle Ω1 ∧ Y.onCircle Ω2 ∧ Y.onCircle Ω3 ∧\n    X ≠ Y := by"}
+{"name": "IMO_2003_P4", "uuid": "f2b8f420-7335-4f8e-bb3e-642f376172a7", "statement_id": "fe7b4ff0-162f-4a34-b65b-c2f020d2d268", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2003_P4 :\n  ∀ (A B C D P Q R X : Point) (BC CA AB : Line),\n    cyclic A B C D ∧\n    distinctPointsOnLine B C BC ∧\n    distinctPointsOnLine C A CA ∧\n    distinctPointsOnLine A B AB ∧\n    foot D P BC ∧\n    foot D Q CA ∧\n    foot D R AB →\n  (|(P─Q)| = |(Q─R)| ↔ ∃ X : Point, X.onLine CA ∧ ∠A:B:X = ∠X:B:C ∧ ∠A:D:X = ∠X:D:C) := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2003_P4 :\n  ∀ (A B C D P Q R X : Point) (BC CA AB : Line),\n    cyclic A B C D ∧\n    distinctPointsOnLine B C BC ∧\n    distinctPointsOnLine C A CA ∧\n    distinctPointsOnLine A B AB ∧\n    foot D P BC ∧\n    foot D Q CA ∧\n    foot D R AB →\n  (|(P─Q)| = |(Q─R)| ↔ ∃ X : Point, X.onLine CA ∧ ∠A:B:X = ∠X:B:C ∧ ∠A:D:X = ∠X:D:C) := by"}
+{"name": "IMO_2008_P1", "uuid": "e7f6e19c-4a39-4ea4-8fae-6c960233630d", "statement_id": "b83400eb-ae59-4c4a-b801-1b6f648ac3d4", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2008_P1 :\n  ∀ (A B C H D E F A1 A2 B1 B2 C1 C2 MBC MCA MAB : Point) (AB BC CA : Line) (ΓA ΓB ΓC : Circle),\n    formAcuteTriangle A B C AB BC CA ∧\n    orthoCentre H A B C D E F AB BC CA ∧\n    midpoint B MBC C ∧\n    MBC.isCentre ΓA ∧\n    H.onCircle ΓA ∧\n    distinctPointsOnLine A1 A2 BC ∧\n    A1.onCircle ΓA ∧\n    A2.onCircle ΓA ∧\n    midpoint C MCA A ∧\n    MCA.isCentre ΓB ∧\n    H.onCircle ΓB ∧\n    distinctPointsOnLine B1 B2 CA ∧\n    B1.onCircle ΓB ∧\n    B2.onCircle ΓB ∧\n    midpoint A MAB B ∧\n    MAB.isCentre ΓC ∧\n    H.onCircle ΓC ∧\n    distinctPointsOnLine C1 C2 AB ∧\n    C1.onCircle ΓC ∧\n    C2.onCircle ΓC\n  → ∃ (Ω : Circle), A1.onCircle Ω ∧ A2.onCircle Ω ∧ B1.onCircle Ω ∧ B2.onCircle Ω ∧ C1.onCircle Ω ∧ C2.onCircle Ω := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2008_P1 :\n  ∀ (A B C H D E F A1 A2 B1 B2 C1 C2 MBC MCA MAB : Point) (AB BC CA : Line) (ΓA ΓB ΓC : Circle),\n    formAcuteTriangle A B C AB BC CA ∧\n    orthoCentre H A B C D E F AB BC CA ∧\n    midpoint B MBC C ∧\n    MBC.isCentre ΓA ∧\n    H.onCircle ΓA ∧\n    distinctPointsOnLine A1 A2 BC ∧\n    A1.onCircle ΓA ∧\n    A2.onCircle ΓA ∧\n    midpoint C MCA A ∧\n    MCA.isCentre ΓB ∧\n    H.onCircle ΓB ∧\n    distinctPointsOnLine B1 B2 CA ∧\n    B1.onCircle ΓB ∧\n    B2.onCircle ΓB ∧\n    midpoint A MAB B ∧\n    MAB.isCentre ΓC ∧\n    H.onCircle ΓC ∧\n    distinctPointsOnLine C1 C2 AB ∧\n    C1.onCircle ΓC ∧\n    C2.onCircle ΓC\n  → ∃ (Ω : Circle), A1.onCircle Ω ∧ A2.onCircle Ω ∧ B1.onCircle Ω ∧ B2.onCircle Ω ∧ C1.onCircle Ω ∧ C2.onCircle Ω := by"}
+{"name": "IMO_2007_P4", "uuid": "30b9f10a-4ff2-4738-9ad4-93cf2c885275", "statement_id": "81555611-4343-43d9-af30-e7d33c4cdd2f", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2007_P4 :\n  ∀ (A B C R P Q K L : Point) (AB BC CA L1 L2 : Line) (Ω : Circle),\n    formTriangle A B C AB BC CA ∧\n    circumCircle Ω A B C ∧\n    R.onCircle Ω ∧\n    R ≠ C ∧\n    ∠ B:C:R = ∠ R:C:A ∧\n    perpBisector B C L1 ∧ P.onLine L1 ∧ ∠ B:C:P = ��� P:C:A ∧\n    perpBisector A C L2 ∧ Q.onLine L2 ∧ ∠ B:C:Q = ∠ Q:C:A ∧\n    midpoint B K C ∧ midpoint A L C\n    → (△ R P K).area = (△ R Q L).area := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2007_P4 :\n  ∀ (A B C R P Q K L : Point) (AB BC CA L1 L2 : Line) (Ω : Circle),\n    formTriangle A B C AB BC CA ∧\n    circumCircle Ω A B C ∧\n    R.onCircle Ω ∧\n    R ≠ C ∧\n    ∠ B:C:R = ∠ R:C:A ∧\n    perpBisector B C L1 ∧ P.onLine L1 ∧ ∠ B:C:P = ∠ P:C:A ∧\n    perpBisector A C L2 ∧ Q.onLine L2 ∧ ∠ B:C:Q = ∠ Q:C:A ∧\n    midpoint B K C ∧ midpoint A L C\n    → (△ R P K).area = (△ R Q L).area := by"}
+{"name": "IMO_2019_P6", "uuid": "eb0c7284-9877-4f1b-b5b2-bd642e04bcec", "statement_id": "894d1421-0759-494b-b9f9-d6d49c4433c6", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2019_P6 :\n  ∀ (A B C I D E F R P Q X : Point)\n    (ω : Circle)\n    (BC CA AB EF Ld ARline AIline DIline PQline Lperp : Line),\n    formAcuteTriangle A B C AB BC CA ∧\n    |(A─B)| ≠ |(A─C)| ∧\n    inCircle ω AB BC CA ∧\n    inCentre I A B C ∧\n    tangentAtPoint D I BC ω ∧\n    tangentAtPoint E I CA ω ∧\n    tangentAtPoint F I AB ω ∧\n    distinctPointsOnLine E F EF ∧\n    distinctPointsOnLine D R Ld ∧\n    perpLineatPoint Ld EF D ∧\n    R.onCircle ω ∧\n    distinctPointsOnLine A R ARline ∧\n    P.onLine ARline ∧\n    P.onCircle ω ∧\n    P ≠ R ∧\n    cyclic P C E Q ∧\n    cyclic P B F Q ∧\n    Q ≠ P ∧\n    distinctPointsOnLine A I AIline ∧\n    distinctPointsOnLine D I DIline ∧\n    distinctPointsOnLine P Q PQline ∧\n    twoLinesIntersectAtPoint DIline PQline X ∧\n    perpLineatPoint Lperp AIline A\n    → X.onLine Lperp :=\n  by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2019_P6 :\n  ∀ (A B C I D E F R P Q X : Point)\n    (ω : Circle)\n    (BC CA AB EF Ld ARline AIline DIline PQline Lperp : Line),\n    formAcuteTriangle A B C AB BC CA ∧\n    |(A─B)| ≠ |(A─C)| ∧\n    inCircle ω AB BC CA ∧\n    inCentre I A B C ∧\n    tangentAtPoint D I BC ω ∧\n    tangentAtPoint E I CA ω ∧\n    tangentAtPoint F I AB ω ∧\n    distinctPointsOnLine E F EF ∧\n    distinctPointsOnLine D R Ld ∧\n    perpLineatPoint Ld EF D ∧\n    R.onCircle ω ∧\n    distinctPointsOnLine A R ARline ∧\n    P.onLine ARline ∧\n    P.onCircle ω ∧\n    P ≠ R ∧\n    cyclic P C E Q ∧\n    cyclic P B F Q ∧\n    Q ≠ P ∧\n    distinctPointsOnLine A I AIline ∧\n    distinctPointsOnLine D I DIline ∧\n    distinctPointsOnLine P Q PQline ∧\n    twoLinesIntersectAtPoint DIline PQline X ∧\n    perpLineatPoint Lperp AIline A\n    → X.onLine Lperp :=\n  by"}
+{"name": "my_favorite_theorem (n", "uuid": "f46ecbd1-f4e8-4823-a3fc-1c81bb305141", "statement_id": "07c0f7dc-34e9-41f8-aaf8-5e936b30bcb3", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem my_favorite_theorem (n : ℕ) (hn : 3 ≤ n) (C : Fin n → (EuclideanSpace ℝ (Fin 2)))\n    (hC : ∀ i, C i ∈ Metric.sphere 0 1) (hC1 : ∀ i j, i ≠ j → Disjoint {x | ∃ t : ℝ, x = C i + t • (C j - C i)} {x | ∃ u : ℝ, x = C j + u • (C i - C j)}) :\n    ∑ i, ∑ j, (if i < j then (1 / dist (C i) (C j)) else 0) ≤ ((n - 1) * π) / 4 := by sorry\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem my_favorite_theorem (n : ℕ) (hn : 3 ≤ n) (C : Fin n → (EuclideanSpace ℝ (Fin 2)))\n    (hC : ∀ i, C i ∈ Metric.sphere 0 1) (hC1 : ∀ i j, i ≠ j → Disjoint {x | ∃ t : ℝ, x = C i + t • (C j - C i)} {x | ∃ u : ℝ, x = C j + u • (C i - C j)}) :\n    ∑ i, ∑ j, (if i < j then (1 / dist (C i) (C j)) else 0) ≤ ((n - 1) * π) / 4 := by sorry"}
+{"name": "IMO_2011_P6", "uuid": "042de375-07ef-453f-ac7b-3ed4ecfa3a78", "statement_id": "cc42e65b-2b7e-4ab3-ac31-92828cac9238", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2011_P6 :\n  ∀ (A B C A' B' C' P O : Point) (ℓ ℓ_a ℓ_b ℓ_c : Line) (Γ Δ : Circle),\n    acuteTriangle A B C ∧\n    circumCircle Γ A B C ∧\n    O.isCentre Γ ∧\n    tangentAtPoint P O ℓ Γ ∧\n    reflectLine ℓ BC ℓ_a ∧\n    reflectLine ℓ CA ℓ_b ∧\n    reflectLine ℓ AB ℓ_c ∧\n    twoLinesIntersectAtPoint ℓ_b ℓ_c A' ∧\n    twoLinesIntersectAtPoint ℓ_c ℓ_a B' ∧\n    twoLinesIntersectAtPoint ℓ_a ℓ_b C' ∧\n    circumCircle Δ A' B' C' →\n    tangentCircles Γ Δ := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2011_P6 :\n  ∀ (A B C A' B' C' P O : Point) (ℓ ℓ_a ℓ_b ℓ_c : Line) (Γ Δ : Circle),\n    acuteTriangle A B C ∧\n    circumCircle Γ A B C ∧\n    O.isCentre Γ ∧\n    tangentAtPoint P O ℓ Γ ∧\n    reflectLine ℓ BC ℓ_a ∧\n    reflectLine ℓ CA ℓ_b ∧\n    reflectLine ℓ AB ℓ_c ∧\n    twoLinesIntersectAtPoint ℓ_b ℓ_c A' ∧\n    twoLinesIntersectAtPoint ℓ_c ℓ_a B' ∧\n    twoLinesIntersectAtPoint ℓ_a ℓ_b C' ∧\n    circumCircle Δ A' B' C' →\n    tangentCircles Γ Δ := by"}
+{"name": "IMO_2021_P4", "uuid": "aa15a578-18a2-4df8-a152-50ae8e18ed5d", "statement_id": "e787b496-e2fd-4e8b-91bd-729114828026", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2021_P4 :\n  ∀ (A B C D I X Y Z T : Point) (Γ Ω : Circle) (AB BC CD DA : Line),\n    formQuadrilateral A B C D AB BC CD DA ∧\n    I.isCentre Γ ∧\n    tangentLine AB Γ ∧ tangentLine BC Γ ∧ tangentLine CD Γ ∧ tangentLine DA Γ ∧\n    circumCircle Ω A I C ∧\n    X.onCircle Ω ∧ between B A X ∧\n    Z.onCircle Ω ∧ between B C Z ∧\n    Y.onCircle Ω ∧ between A D Y ∧\n    T.onCircle Ω ∧ between C D T →\n    |(A─D)| + |(D─T)| + |(T─X)| + |(X─A)| =\n    |(C─D)| + |(D─Y)| + |(Y─Z)| + |(Z─C)| := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2021_P4 :\n  ∀ (A B C D I X Y Z T : Point) (Γ Ω : Circle) (AB BC CD DA : Line),\n    formQuadrilateral A B C D AB BC CD DA ∧\n    I.isCentre Γ ∧\n    tangentLine AB Γ ∧ tangentLine BC Γ ∧ tangentLine CD Γ ∧ tangentLine DA Γ ∧\n    circumCircle Ω A I C ∧\n    X.onCircle Ω ∧ between B A X ∧\n    Z.onCircle Ω ∧ between B C Z ∧\n    Y.onCircle Ω ∧ between A D Y ∧\n    T.onCircle Ω ∧ between C D T →\n    |(A─D)| + |(D─T)| + |(T─X)| + |(X─A)| =\n    |(C─D)| + |(D─Y)| + |(Y─Z)| + |(Z─C)| := by"}
+{"name": "imo2013_p3", "uuid": "f638b703-0a0f-47d4-84ac-67190d5b63f1", "statement_id": "04049b09-3725-4ce6-9071-d7b3ee4c85e3", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem imo2013_p3 :\n  ∀ (A B C Ia Ib Ic A1 B1 C1 O1 : Point)\n    (exA exB exC Ω : Circle)\n    (AB BC CA : Line),\n    formTriangle A B C AB BC CA ∧\n    exCentre Ia A B C ∧ exCentre Ib B C A ∧ exCentre Ic C A B ∧\n    Ia.isCentre exA ∧ tangentAtPoint A1 Ia BC exA ∧ between B A1 C ∧\n    Ib.isCentre exB ∧ tangentAtPoint B1 Ib CA exB ∧ between C B1 A ∧\n    Ic.isCentre exC ∧ tangentAtPoint C1 Ic AB exC ∧ between A C1 B ∧\n    circumCentre O1 A1 B1 C1 ∧ circumCircle Ω A B C ∧\n    O1.onCircle Ω\n    → rightTriangle A B C := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem imo2013_p3 :\n  ∀ (A B C Ia Ib Ic A1 B1 C1 O1 : Point)\n    (exA exB exC Ω : Circle)\n    (AB BC CA : Line),\n    formTriangle A B C AB BC CA ∧\n    exCentre Ia A B C ∧ exCentre Ib B C A ∧ exCentre Ic C A B ∧\n    Ia.isCentre exA ∧ tangentAtPoint A1 Ia BC exA ∧ between B A1 C ∧\n    Ib.isCentre exB ∧ tangentAtPoint B1 Ib CA exB ∧ between C B1 A ∧\n    Ic.isCentre exC ∧ tangentAtPoint C1 Ic AB exC ∧ between A C1 B ∧\n    circumCentre O1 A1 B1 C1 ∧ circumCircle Ω A B C ∧\n    O1.onCircle Ω\n    → rightTriangle A B C := by"}
+{"name": "IMO_2012_P5", "uuid": "213f114c-b8f3-4de0-b632-2e76ca51942c", "statement_id": "e9819d07-95cd-4809-989a-ced42cd02e87", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2012_P5 :\n  ∀ (A B C D X K L M : Point) (AB BC CA AL BK : Line),\n    formTriangle A B C AB BC CA ∧ ∠ B:C:A = ∟ ∧ foot C D AB ∧\n    between C X D ∧ between A K X ∧ |(B─K)| = |(B─C)| ∧\n    between B L X ∧ |(A─L)| = |(A─C)| ∧\n    distinctPointsOnLine A L AL ∧ distinctPointsOnLine B K BK ∧\n    M.onLine AL ∧ M.onLine BK →\n    |(M─K)| = |(M─L)| := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2012_P5 :\n  ∀ (A B C D X K L M : Point) (AB BC CA AL BK : Line),\n    formTriangle A B C AB BC CA ∧ ∠ B:C:A = ∟ ∧ foot C D AB ∧\n    between C X D ∧ between A K X ∧ |(B─K)| = |(B─C)| ∧\n    between B L X ∧ |(A─L)| = |(A─C)| ∧\n    distinctPointsOnLine A L AL ∧ distinctPointsOnLine B K BK ∧\n    M.onLine AL ∧ M.onLine BK →\n    |(M─K)| = |(M─L)| := by"}
+{"name": "IMO_2019_P2", "uuid": "f610dd85-40fe-4871-9e7e-b4e8414bb9ac", "statement_id": "848d7a4e-36c7-4774-a263-b096bd3f0bb4", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2019_P2 :\n  ∀ (A B C A1 B1 P Q P1 Q1 : Point)\n    (BC AC AA1 BB1 PQ AB PB1 QA1 : Line),\n    distinctPointsOnLine B C BC ∧\n    distinctPointsOnLine A C AC ∧\n    distinctPointsOnLine A A1 AA1 ∧\n    distinctPointsOnLine B B1 BB1 ∧\n    between B A1 C ∧\n    between A B1 C ∧\n    between A P A1 ∧\n    between B Q B1 ∧\n    distinctPointsOnLine P Q PQ ∧\n    distinctPointsOnLine A B AB ∧\n    ¬ AB.intersectsLine PQ ∧\n    distinctPointsOnLine P B1 PB1 ∧\n    P1.onLine PB1 ∧\n    between P B1 P1 ∧\n    distinctPointsOnLine Q A1 QA1 ∧\n    Q1.onLine QA1 ∧\n    between Q A1 Q1 ∧\n    ∠ P:P1:C = ∠ B:A:C ∧\n    ∠ C:Q1:Q = ∠ C:B:A →\n    cyclic P Q P1 Q1 := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2019_P2 :\n  ∀ (A B C A1 B1 P Q P1 Q1 : Point)\n    (BC AC AA1 BB1 PQ AB PB1 QA1 : Line),\n    distinctPointsOnLine B C BC ∧\n    distinctPointsOnLine A C AC ∧\n    distinctPointsOnLine A A1 AA1 ∧\n    distinctPointsOnLine B B1 BB1 ∧\n    between B A1 C ∧\n    between A B1 C ∧\n    between A P A1 ∧\n    between B Q B1 ∧\n    distinctPointsOnLine P Q PQ ∧\n    distinctPointsOnLine A B AB ∧\n    ¬ AB.intersectsLine PQ ∧\n    distinctPointsOnLine P B1 PB1 ∧\n    P1.onLine PB1 ∧\n    between P B1 P1 ∧\n    distinctPointsOnLine Q A1 QA1 ∧\n    Q1.onLine QA1 ∧\n    between Q A1 Q1 ∧\n    ∠ P:P1:C = ∠ B:A:C ∧\n    ∠ C:Q1:Q = ∠ C:B:A →\n    cyclic P Q P1 Q1 := by"}
+{"name": "IMO_2010_P2", "uuid": "8c5cfa34-6ded-4a8c-b682-409c4835012c", "statement_id": "cc08b01b-2d14-4221-b16e-83e076c3574c", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2010_P2 :\n  ∀ (A B C D E F G I X : Point) (Γ : Circle) (AB BC CA AI EI DG : Line),\n    distinctPointsOnLine A B AB ∧\n    distinctPointsOnLine B C BC ∧\n    distinctPointsOnLine C A CA ∧\n    formTriangle A B C AB BC CA ∧\n    inCentre I A B C ∧\n    circumCircle Γ A B C ∧\n    distinctPointsOnLine A I AI ∧\n    D.onCircle Γ ∧\n    D.onLine AI ∧\n    D ≠ A ∧\n    distinctPointsOnLine E I EI ∧\n    E.onCircle Γ ∧\n    E.opposingSides A E BC ∧\n    between B F C ∧\n    ∠ B:A:F = ∠ C:A:E ∧\n    (∠ B:A:F + ∠ B:A:F < ∠ B:A:C) ∧\n    midpoint I G F ∧\n    distinctPointsOnLine D G DG ∧\n    X.onLine EI ∧\n    X.onLine DG →\n    X.onCircle Γ := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2010_P2 :\n  ∀ (A B C D E F G I X : Point) (Γ : Circle) (AB BC CA AI EI DG : Line),\n    distinctPointsOnLine A B AB ∧\n    distinctPointsOnLine B C BC ∧\n    distinctPointsOnLine C A CA ∧\n    formTriangle A B C AB BC CA ∧\n    inCentre I A B C ∧\n    circumCircle Γ A B C ∧\n    distinctPointsOnLine A I AI ∧\n    D.onCircle Γ ∧\n    D.onLine AI ∧\n    D ≠ A ∧\n    distinctPointsOnLine E I EI ∧\n    E.onCircle Γ ∧\n    E.opposingSides A E BC ∧\n    between B F C ∧\n    ∠ B:A:F = ∠ C:A:E ∧\n    (∠ B:A:F + ∠ B:A:F < ∠ B:A:C) ∧\n    midpoint I G F ∧\n    distinctPointsOnLine D G DG ∧\n    X.onLine EI ∧\n    X.onLine DG →\n    X.onCircle Γ := by"}
+{"name": "IMO_2009_P2", "uuid": "6600ebc0-09b1-4957-ab3d-ffbf8ebe8452", "statement_id": "ba9434dd-27d4-4d95-bfd2-22709fc56209", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2009_P2 :\n  ∀ (A B C O P Q K L M OΓ : Point) (AB BC CA PQ : Line) (Γ : Circle),\n    formTriangle A B C AB BC CA ∧\n    circumCentre O A B C ∧\n    P.onLine CA ∧ between C P A ∧\n    Q.onLine AB ∧ between A Q B ∧\n    midpoint B K P ∧\n    midpoint C L Q ∧\n    midpoint P M Q ∧\n    circumCircle Γ K L M ∧\n    tangentAtPoint M OΓ PQ Γ ��\n    |(O─P)| = |(O─Q)| := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2009_P2 :\n  ∀ (A B C O P Q K L M OΓ : Point) (AB BC CA PQ : Line) (Γ : Circle),\n    formTriangle A B C AB BC CA ∧\n    circumCentre O A B C ∧\n    P.onLine CA ∧ between C P A ∧\n    Q.onLine AB ∧ between A Q B ∧\n    midpoint B K P ∧\n    midpoint C L Q ∧\n    midpoint P M Q ∧\n    circumCircle Γ K L M ∧\n    tangentAtPoint M OΓ PQ Γ →\n    |(O─P)| = |(O─Q)| := by"}
+{"name": "IMO_2020_P1", "uuid": "99a6c37d-75ec-4181-8ad4-dc0d43a93aa6", "statement_id": "993dee02-9fa6-43fb-9e0b-ec0d28a01b4f", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2020_P1 :\n  ∀ (A B C D P : Point) (AB BC CD DA l1 l2 l3 : Line),\n    formQuadrilateral A B C D AB BC CD DA ∧\n    insideQuadrilateral P A B C D AB BC CD DA ∧\n    ∠ P:B:A = 2 * ∠ P:A:D ∧\n    ∠ D:P:A = 3 * ∠ P:A:D ∧\n    ∠ B:A:P = 2 * ∠ C:B:P ∧\n    ∠ B:P:C = 3 * ∠ C:B:P ∧\n    D.onLine l1 ∧\n    (∀ (X : Point), X.onLine l1 → ∠ A:D:X = ∠ P:D:X) ∧\n    A.opposingSides P l1 ∧\n    C.onLine l2 ∧\n    (∀ (X : Point), X.onLine l2 → ∠ P:C:X = ∠ B:C:X) ∧\n    P.opposingSides B l2 ∧\n    perpBisector A B l3 →\n  concurrent l1 l2 l3 := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2020_P1 :\n  ∀ (A B C D P : Point) (AB BC CD DA l1 l2 l3 : Line),\n    formQuadrilateral A B C D AB BC CD DA ∧\n    insideQuadrilateral P A B C D AB BC CD DA ∧\n    ∠ P:B:A = 2 * ∠ P:A:D ∧\n    ∠ D:P:A = 3 * ∠ P:A:D ∧\n    ∠ B:A:P = 2 * ∠ C:B:P ∧\n    ∠ B:P:C = 3 * ∠ C:B:P ∧\n    D.onLine l1 ∧\n    (∀ (X : Point), X.onLine l1 → ∠ A:D:X = ∠ P:D:X) ∧\n    A.opposingSides P l1 ∧\n    C.onLine l2 ∧\n    (∀ (X : Point), X.onLine l2 → ∠ P:C:X = ∠ B:C:X) ∧\n    P.opposingSides B l2 ∧\n    perpBisector A B l3 →\n  concurrent l1 l2 l3 := by"}
+{"name": "IMO_2015_P4", "uuid": "eb700e2f-826f-4a41-bb12-45973905fc6e", "statement_id": "987c9531-0a93-443b-814f-1da38e46e370", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2015_P4 :\n  ∀ (A B C O D E F G K L X : Point) (BC AB CA : Line) (Ω Γ : Circle),\n    formTriangle A B C AB BC CA ∧\n    circumCentre O A B C ∧\n    circumCircle Ω A B C ∧\n    O.isCentre Ω ∧\n    A.isCentre Γ ∧\n    distinctPointsOnLine B C BC ∧\n    D.onLine BC ∧ E.onLine BC ∧\n    between B D C ∧ between B E C ∧ D ≠ E ∧\n    circlesIntersectsAtTwoPoints Γ Ω F G ∧\n    cyclic B D F K ∧ between A K B ∧\n    cyclic C G E L ∧ between C L A ∧\n    ¬ coll F G K ∧\n    coll F K X ∧ coll G L X\n    → coll A O X := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2015_P4 :\n  ∀ (A B C O D E F G K L X : Point) (BC AB CA : Line) (Ω Γ : Circle),\n    formTriangle A B C AB BC CA ∧\n    circumCentre O A B C ∧\n    circumCircle Ω A B C ∧\n    O.isCentre Ω ∧\n    A.isCentre Γ ∧\n    distinctPointsOnLine B C BC ∧\n    D.onLine BC ∧ E.onLine BC ∧\n    between B D C ∧ between B E C ∧ D ≠ E ∧\n    circlesIntersectsAtTwoPoints Γ Ω F G ∧\n    cyclic B D F K ∧ between A K B ∧\n    cyclic C G E L ∧ between C L A ∧\n    ¬ coll F G K ∧\n    coll F K X ∧ coll G L X\n    → coll A O X := by"}
+{"name": "IMO_2024_P4", "uuid": "cda11974-ad1e-4a3b-9ae9-d1e9b7049cf7", "statement_id": "35c63e6c-6642-4afa-86df-ee5ac70a8c82", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2024_P4 :\n  ∀ (A B C I X Y P K L : Point) (AB BC AC AI LX LY : Line) (Ω Γ : Circle),\n    formTriangle A B C AB BC AC ∧\n    |(A─B)| < |(A─C)| ∧\n    |(A─C)| < |(B─C)| ∧\n    inCentre I A B C ∧\n    I.isCentre Ω ∧\n    inCircle Ω AB BC AC ∧\n    distinctPointsOnLine X C BC ∧\n    X.onLine LX ∧\n    ¬ LX.intersectsLine AC ∧\n    tangentLine LX Ω ∧\n    distinctPointsOnLine Y B BC ∧\n    Y.onLine LY ∧\n    ¬ LY.intersectsLine AB ∧\n    tangentLine LY Ω ∧\n    circumCircle Γ A B C ∧\n    P.onLine AI ∧\n    P.onCircle Γ ∧\n    P ≠ A ∧\n    midpoint A K C ∧\n    midpoint A L B →\n  ∠ K:I:L + ∠ Y:P:X = ∟ + ∟ := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2024_P4 :\n  ∀ (A B C I X Y P K L : Point) (AB BC AC AI LX LY : Line) (Ω Γ : Circle),\n    formTriangle A B C AB BC AC ∧\n    |(A─B)| < |(A─C)| ∧\n    |(A─C)| < |(B─C)| ∧\n    inCentre I A B C ∧\n    I.isCentre Ω ∧\n    inCircle Ω AB BC AC ∧\n    distinctPointsOnLine X C BC ∧\n    X.onLine LX ∧\n    ¬ LX.intersectsLine AC ∧\n    tangentLine LX Ω ∧\n    distinctPointsOnLine Y B BC ∧\n    Y.onLine LY ∧\n    ¬ LY.intersectsLine AB ∧\n    tangentLine LY Ω ∧\n    circumCircle Γ A B C ∧\n    P.onLine AI ∧\n    P.onCircle Γ ∧\n    P ≠ A ∧\n    midpoint A K C ∧\n    midpoint A L B →\n  ∠ K:I:L + ∠ Y:P:X = ∟ + ∟ := by"}
+{"name": "IMO_2012_P1", "uuid": "acdfac59-943f-4177-a843-08b7299ed1db", "statement_id": "8b87dac1-e53a-4a4d-8e8e-8eb96834c37b", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2012_P1 :\n  ∀ (A B C J K L M F G S T : Point)\n    (AB BC CA LM BJ KM CJ AF AG : Line)\n    (Ω : Circle),\n    formTriangle A B C AB BC CA ∧\n    exCentre J A B C ∧\n    J.isCentre Ω ∧\n    tangentAtPoint M J BC Ω ∧\n    tangentAtPoint K J AB Ω ∧\n    tangentAtPoint L J AC Ω ∧\n    distinctPointsOnLine L M LM ∧\n    distinctPointsOnLine B J BJ ∧\n    F.onLine LM ∧\n    F.onLine BJ ∧\n    distinctPointsOnLine K M KM ∧\n    distinctPointsOnLine C J CJ ∧\n    G.onLine KM ∧\n    G.onLine CJ ∧\n    distinctPointsOnLine A F AF ∧\n    S.onLine AF ∧\n    S.onLine BC ∧\n    distinctPointsOnLine A G AG ∧\n    T.onLine AG ∧\n    T.onLine BC\n    → midpoint S M T := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2012_P1 :\n  ∀ (A B C J K L M F G S T : Point)\n    (AB BC CA LM BJ KM CJ AF AG : Line)\n    (Ω : Circle),\n    formTriangle A B C AB BC CA ∧\n    exCentre J A B C ∧\n    J.isCentre Ω ∧\n    tangentAtPoint M J BC Ω ∧\n    tangentAtPoint K J AB Ω ∧\n    tangentAtPoint L J AC Ω ∧\n    distinctPointsOnLine L M LM ∧\n    distinctPointsOnLine B J BJ ∧\n    F.onLine LM ∧\n    F.onLine BJ ∧\n    distinctPointsOnLine K M KM ∧\n    distinctPointsOnLine C J CJ ∧\n    G.onLine KM ∧\n    G.onLine CJ ∧\n    distinctPointsOnLine A F AF ∧\n    S.onLine AF ∧\n    S.onLine BC ∧\n    distinctPointsOnLine A G AG ∧\n    T.onLine AG ∧\n    T.onLine BC\n    → midpoint S M T := by"}
+{"name": "IMO_2007_P2", "uuid": "d681ebb0-efa2-42fd-83df-0f9a0d4aaa94", "statement_id": "45d2c2f2-afff-45c0-b20f-28cef93f1448", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2007_P2 :\n  ∀ (A B C D E F G : Point) (AB BC CD DA l : Line),\n    parallelogram A B C D AB BC CD DA ∧\n    cyclic B C E D ∧\n    A.onLine l ∧\n    between D F C ∧ F.onLine l ∧\n    G.onLine BC ∧ G.onLine l ∧\n    |(E─F)| = |(E─G)| ∧ |(E─G)| = |(E─C)| →\n    ∠ D:A:G = ∠ G:A:B := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2007_P2 :\n  ∀ (A B C D E F G : Point) (AB BC CD DA l : Line),\n    parallelogram A B C D AB BC CD DA ∧\n    cyclic B C E D ∧\n    A.onLine l ∧\n    between D F C ∧ F.onLine l ∧\n    G.onLine BC ∧ G.onLine l ∧\n    |(E─F)| = |(E─G)| ∧ |(E─G)| = |(E─C)| →\n    ∠ D:A:G = ∠ G:A:B := by"}
+{"name": "IMO_2018_P1", "uuid": "0312102d-5da8-4506-84d2-4da1f22d5f1a", "statement_id": "a8300d90-99ab-4684-8a3a-3a555b7304f3", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2018_P1 :\n  ∀ (A B C D E F G : Point)\n    (AB BC CA L1 L2 DE FG : Line)\n    (Γ : Circle),\n    formAcuteTriangle A B C AB BC CA ∧\n    circumCircle Γ A B C ∧\n    between A D B ∧\n    between A E C ∧\n    |(A─D)| = |(A─E)| ∧\n    perpBisector B D L1 ∧\n    F.onCircle Γ ∧\n    F.onLine L1 ∧\n    F.opposingSides C AB ∧\n    perpBisector C E L2 ∧\n    G.onCircle Γ ∧\n    G.onLine L2 ∧\n    G.opposingSides B AC ∧\n    distinctPointsOnLine D E DE ∧\n    distinctPointsOnLine F G FG →\n    ¬(DE.intersectsLine FG) ∨ (coll D E F ∧ coll D E G) := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2018_P1 :\n  ∀ (A B C D E F G : Point)\n    (AB BC CA L1 L2 DE FG : Line)\n    (Γ : Circle),\n    formAcuteTriangle A B C AB BC CA ∧\n    circumCircle Γ A B C ∧\n    between A D B ∧\n    between A E C ∧\n    |(A─D)| = |(A─E)| ∧\n    perpBisector B D L1 ∧\n    F.onCircle Γ ∧\n    F.onLine L1 ∧\n    F.opposingSides C AB ∧\n    perpBisector C E L2 ∧\n    G.onCircle Γ ∧\n    G.onLine L2 ∧\n    G.opposingSides B AC ∧\n    distinctPointsOnLine D E DE ∧\n    distinctPointsOnLine F G FG →\n    ¬(DE.intersectsLine FG) ∨ (coll D E F ∧ coll D E G) := by"}
+{"name": "IMO_2014_P4", "uuid": "31e17362-55e8-4a7d-a56d-0bb6ef67e00c", "statement_id": "97766354-aa4e-42bf-a89a-bfbb9bf3b560", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2014_P4 :\n  ∀ (A B C P Q M N X : Point) (AB BC CA BM CN : Line) (Ω : Circle),\n    formAcuteTriangle A B C AB BC CA ∧\n    between B P C ∧ between B Q C ∧\n    ∠ P:A:B = ∠ B:C:A ∧ ∠ C:A:Q = ∠ A:B:C ∧\n    midpoint A P M ∧ midpoint A Q N ∧\n    distinctPointsOnLine B M BM ∧ distinctPointsOnLine C N CN ∧\n    twoLinesIntersectAtPoint BM CN X ∧\n    circumCircle Ω A B C\n    → X.onCircle Ω := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2014_P4 :\n  ∀ (A B C P Q M N X : Point) (AB BC CA BM CN : Line) (Ω : Circle),\n    formAcuteTriangle A B C AB BC CA ∧\n    between B P C ∧ between B Q C ∧\n    ∠ P:A:B = ∠ B:C:A ∧ ∠ C:A:Q = ∠ A:B:C ∧\n    midpoint A P M ∧ midpoint A Q N ∧\n    distinctPointsOnLine B M BM ∧ distinctPointsOnLine C N CN ∧\n    twoLinesIntersectAtPoint BM CN X ∧\n    circumCircle Ω A B C\n    → X.onCircle Ω := by"}
+{"name": "IMO_2017_P4", "uuid": "2f91c0be-a950-449b-8cca-8397b945338f", "statement_id": "77e78a06-099d-4e49-a802-ebe66b471668", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2017_P4 :\n  ∀ (R S T J A B K O P : Point) (Ω Γ : Circle) (ℓ AJline KTline : Line),\n    R.onCircle Ω ∧\n    S.onCircle Ω ∧\n    R ≠ S ∧\n    tangentAtPoint R O ℓ Ω ∧\n    midpoint R S T ∧\n    J.onCircle Ω ∧\n    J ≠ R ∧\n    J ≠ S ∧\n    circumCircle Γ J S T ∧\n    A.onLine ℓ ∧\n    A.onCircle Γ ∧\n    B.onLine ℓ ∧\n    B.onCircle Γ ∧\n    A ≠ B ∧\n    |(R─A)| < |(R─B)| ∧\n    coll A J K ∧\n    K.onCircle Ω ∧\n    K ≠ J →\n  tangentAtPoint T P KTline Γ := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2017_P4 :\n  ∀ (R S T J A B K O P : Point) (Ω Γ : Circle) (ℓ AJline KTline : Line),\n    R.onCircle Ω ∧\n    S.onCircle Ω ∧\n    R ≠ S ∧\n    tangentAtPoint R O ℓ Ω ∧\n    midpoint R S T ∧\n    J.onCircle Ω ∧\n    J ≠ R ∧\n    J ≠ S ∧\n    circumCircle Γ J S T ∧\n    A.onLine ℓ ∧\n    A.onCircle Γ ∧\n    B.onLine ℓ ∧\n    B.onCircle Γ ∧\n    A ≠ B ∧\n    |(R─A)| < |(R─B)| ∧\n    coll A J K ∧\n    K.onCircle Ω ∧\n    K ≠ J →\n  tangentAtPoint T P KTline Γ := by"}
+{"name": "IMO_2005_P1", "uuid": "c3858742-0abb-46ba-bdbd-f552dc3cfebf", "statement_id": "de10ee1a-6f74-4128-bcae-596b163add7b", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2005_P1 :\n  ∀ (A B C A1 A2 B1 B2 C1 C2 : Point)\n    (AB BC CA A1B2 B1C2 C1A2 : Line),\n    formTriangle A B C AB BC CA ∧\n    |(A─B)| = |(B─C)| ∧ |(B─C)| = |(C─A)| ∧\n    distinctPointsOnLine A1 A2 BC ∧\n    distinctPointsOnLine B1 B2 CA ∧\n    distinctPointsOnLine C1 C2 AB ∧\n    distinctPointsOnLine A1 B2 A1B2 ∧\n    distinctPointsOnLine B1 C2 B1C2 ∧\n    distinctPointsOnLine C1 A2 C1A2 ∧\n    |(A1─A2)| = |(A2─B1)| ∧\n    |(A2─B1)| = |(B1─B2)| ∧\n    |(B1─B2)| = |(B2─C1)| ∧\n    |(B2─C1)| = |(C1─C2)| ∧\n    |(C1─C2)| = |(C2─A1)| →\n    concurrent A1B2 B1C2 C1A2 := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2005_P1 :\n  ∀ (A B C A1 A2 B1 B2 C1 C2 : Point)\n    (AB BC CA A1B2 B1C2 C1A2 : Line),\n    formTriangle A B C AB BC CA ∧\n    |(A─B)| = |(B─C)| ∧ |(B─C)| = |(C─A)| ∧\n    distinctPointsOnLine A1 A2 BC ∧\n    distinctPointsOnLine B1 B2 CA ∧\n    distinctPointsOnLine C1 C2 AB ∧\n    distinctPointsOnLine A1 B2 A1B2 ∧\n    distinctPointsOnLine B1 C2 B1C2 ∧\n    distinctPointsOnLine C1 A2 C1A2 ∧\n    |(A1─A2)| = |(A2─B1)| ∧\n    |(A2─B1)| = |(B1─B2)| ∧\n    |(B1─B2)| = |(B2─C1)| ∧\n    |(B2─C1)| = |(C1─C2)| ∧\n    |(C1─C2)| = |(C2─A1)| →\n    concurrent A1B2 B1C2 C1A2 := by"}
+{"name": "IMO_2003_P3", "uuid": "1adb0326-d046-41b8-bed6-0cecd24c123b", "statement_id": "1d26e7d1-f8b2-4bb0-b6e2-3e0dd04c7bba", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2003_P3 :\n  ∀ (A B C D E F M1 M2 M3 M4 M5 M6 : Point)\n    (AB BC CD DE EF FA : Line),\n    distinctPointsOnLine A B AB ∧ distinctPointsOnLine B C BC ∧ distinctPointsOnLine C D CD ∧\n    distinctPointsOnLine D E DE ∧ distinctPointsOnLine E F EF ∧ distinctPointsOnLine F A FA ∧\n    A.sameSide C DE ∧ B.sameSide D EF ∧ C.sameSide E FA ∧\n    D.sameSide F AB ∧ E.sameSide A BC ∧ F.sameSide B CD ∧\n    midpoint A M1 B ∧ midpoint D M2 E ∧ |(M1─M2)| = (√3/2) * (|(A─B)| + |(D─E)|) ∧\n    midpoint B M3 C ∧ midpoint E M4 F ∧ |(M3─M4)| = (√3/2) * (|(B─C)| + |(E─F)|) ∧\n    midpoint C M5 D ∧ midpoint F M6 A ∧ |(M5─M6)| = (√3/2) * (|(C─D)| + |(F─A)|) →\n    ∠ F:A:B = ∠ A:B:C ∧\n    ∠ A:B:C = ∠ B:C:D ∧\n    ∠ B:C:D = ∠ C:D:E ∧\n    ∠ C:D:E = ∠ D:E:F ∧\n    ∠ D:E:F = ∠ E:F:A := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2003_P3 :\n  ∀ (A B C D E F M1 M2 M3 M4 M5 M6 : Point)\n    (AB BC CD DE EF FA : Line),\n    distinctPointsOnLine A B AB ∧ distinctPointsOnLine B C BC ∧ distinctPointsOnLine C D CD ∧\n    distinctPointsOnLine D E DE ∧ distinctPointsOnLine E F EF ∧ distinctPointsOnLine F A FA ∧\n    A.sameSide C DE ∧ B.sameSide D EF ∧ C.sameSide E FA ∧\n    D.sameSide F AB ∧ E.sameSide A BC ∧ F.sameSide B CD ∧\n    midpoint A M1 B ∧ midpoint D M2 E ∧ |(M1─M2)| = (√3/2) * (|(A─B)| + |(D─E)|) ∧\n    midpoint B M3 C ∧ midpoint E M4 F ∧ |(M3─M4)| = (√3/2) * (|(B─C)| + |(E─F)|) ∧\n    midpoint C M5 D ∧ midpoint F M6 A ∧ |(M5─M6)| = (√3/2) * (|(C─D)| + |(F─A)|) →\n    ∠ F:A:B = ∠ A:B:C ∧\n    ∠ A:B:C = ∠ B:C:D ∧\n    ∠ B:C:D = ∠ C:D:E ∧\n    ∠ C:D:E = ∠ D:E:F ∧\n    ∠ D:E:F = ∠ E:F:A := by"}
+{"name": "IMO_2023_P2", "uuid": "c100b6c1-6ac8-4f13-9974-0038e0fc8c20", "statement_id": "345e9b3e-7140-4276-bd89-cd1c2e9a6bf4", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2023_P2 :\n  ∀ (A B C S D E L P M Oω : Point) (Ω ω : Circle)\n    (AB BC CA BS BE DL TL : Line),\n    formAcuteTriangle A B C AB BC CA ∧\n    |(A─B)| < |(A─C)| ∧\n    circumCircle Ω A B C ∧\n    S.onCircle Ω ∧\n    |(C─S)| = |(B─S)| ∧\n    S.sameSide A BC ∧\n    foot A D BC ∧\n    distinctPointsOnLine B S BS ∧\n    D.onLine BS ∧\n    distinctPointsOnLine B E BE ∧\n    E.onCircle Ω ∧\n    coll A D E ∧\n    E ≠ A ∧\n    distinctPointsOnLine D L DL ∧\n    ¬ DL.intersectsLine BC ∧\n    L.onLine BE ∧\n    L.onLine DL ∧\n    circumCircle ω B D L ∧\n    P.onCircle Ω ∧\n    P.onCircle ω ∧\n    P ≠ B ∧\n    Oω.isCentre ω →\n    tangentAtPoint P Oω TL ω ∧\n    M.onLine TL ∧\n    M.onLine BS ∧\n    ∠ B:A:M = ∠ M:A:C := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2023_P2 :\n  ∀ (A B C S D E L P M Oω : Point) (Ω ω : Circle)\n    (AB BC CA BS BE DL TL : Line),\n    formAcuteTriangle A B C AB BC CA ∧\n    |(A─B)| < |(A─C)| ∧\n    circumCircle Ω A B C ∧\n    S.onCircle Ω ∧\n    |(C─S)| = |(B─S)| ∧\n    S.sameSide A BC ∧\n    foot A D BC ∧\n    distinctPointsOnLine B S BS ∧\n    D.onLine BS ∧\n    distinctPointsOnLine B E BE ∧\n    E.onCircle Ω ∧\n    coll A D E ∧\n    E ≠ A ∧\n    distinctPointsOnLine D L DL ∧\n    ¬ DL.intersectsLine BC ∧\n    L.onLine BE ∧\n    L.onLine DL ∧\n    circumCircle ω B D L ∧\n    P.onCircle Ω ∧\n    P.onCircle ω ∧\n    P ≠ B ∧\n    Oω.isCentre ω →\n    tangentAtPoint P Oω TL ω ∧\n    M.onLine TL ∧\n    M.onLine BS ∧\n    ∠ B:A:M = ∠ M:A:C := by"}
+{"name": "IMO_2004_P5", "uuid": "c92622c1-0842-44fc-a395-4e32155ccc13", "statement_id": "b321014c-a7e2-4726-9de2-84e81be6ac7e", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2004_P5 :\n  ∀ (A B C D P : Point) (AB BC CD DA BD : Line),\n    formQuadrilateral A B C D AB BC CD DA ∧\n    ¬ (∠ A:B:D = ∠ D:B:C) ∧\n    ¬ (∠ C:D:B = ∠ B:D:A) ∧\n    insideQuadrilateral P A B C D AB BC CD DA ∧\n    ∠ P:B:C = ∠ D:B:A ∧\n    ∠ P:D:C = ∠ B:D:A →\n    (cyclic A B C D ↔ |(A─P)| = |(C─P)|) := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2004_P5 :\n  ∀ (A B C D P : Point) (AB BC CD DA BD : Line),\n    formQuadrilateral A B C D AB BC CD DA ∧\n    ¬ (∠ A:B:D = ∠ D:B:C) ∧\n    ¬ (∠ C:D:B = ∠ B:D:A) ∧\n    insideQuadrilateral P A B C D AB BC CD DA ∧\n    ∠ P:B:C = ∠ D:B:A ∧\n    ∠ P:D:C = ∠ B:D:A →\n    (cyclic A B C D ↔ |(A─P)| = |(C─P)|) := by"}
+{"name": "IMO_2021_P3", "uuid": "507aa2f1-d82f-44a5-a4b9-1027bb6b9ee0", "statement_id": "9e2a3f8c-e02f-4e16-884a-acca4e008ce1", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2021_P3 :\n  ∀ (A B C D E F X O1 O2 : Point) (AB BC CA EF L12 : Line),\n    formAcuteTriangle A B C AB BC CA ∧\n    insideTriangle D A B C AB BC CA ∧\n    |(A─B)| > |(A─C)| ∧\n    ∠ D:A:B = ∠ C:A:D ∧\n    between A E C ∧\n    ∠ A:D:E = ∠ B:C:D ∧\n    between A F B ∧\n    ∠ F:D:A = ∠ D:B:C ∧\n    X.onLine CA ∧\n    |(C─X)| = |(B─X)| ∧\n    circumCentre O1 A D C ∧\n    circumCentre O2 E X D ∧\n    distinctPointsOnLine E F EF ∧\n    distinctPointsOnLine O1 O2 L12 →\n    concurrent BC EF L12 := by sorry\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2021_P3 :\n  ∀ (A B C D E F X O1 O2 : Point) (AB BC CA EF L12 : Line),\n    formAcuteTriangle A B C AB BC CA ∧\n    insideTriangle D A B C AB BC CA ∧\n    |(A─B)| > |(A─C)| ∧\n    ∠ D:A:B = ∠ C:A:D ∧\n    between A E C ∧\n    ∠ A:D:E = ∠ B:C:D ∧\n    between A F B ∧\n    ∠ F:D:A = ∠ D:B:C ∧\n    X.onLine CA ∧\n    |(C─X)| = |(B─X)| ∧\n    circumCentre O1 A D C ∧\n    circumCentre O2 E X D ∧\n    distinctPointsOnLine E F EF ∧\n    distinctPointsOnLine O1 O2 L12 →\n    concurrent BC EF L12 := by sorry"}
+{"name": "IMO_2000_P6", "uuid": "3fecf0a0-3bc6-4273-9c97-657f3b26e44d", "statement_id": "16e08fa2-d39b-4fa4-ab7d-d88bf57546b3", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2000_P6 :\n  ∀ (A B C H1 H2 H3 T1 T2 T3 X Y Z : Point)\n    (AB BC CA T12 T23 T31 L12 L23 L31 : Line) (Ω : Circle),\n    formAcuteTriangle A B C AB BC CA ∧\n    foot A H1 BC ∧\n    foot B H2 CA ∧\n    foot C H3 AB ∧\n    T1.onLine BC ∧ T1.onCircle Ω ∧\n    T2.onLine CA ∧ T2.onCircle Ω ∧\n    T3.onLine AB ∧ T3.onCircle Ω ∧\n    distinctPointsOnLine T1 T2 T12 ∧\n    distinctPointsOnLine T2 T3 T23 ∧\n    distinctPointsOnLine T3 T1 T31 ∧\n    symmetricLine H1H2 T12 L12 ∧\n    symmetricLine H2H3 T23 L23 ∧\n    symmetricLine H3H1 T31 L31 →\n    twoLinesIntersectAtPoint L12 L23 X ∧ X.onCircle Ω ∧\n    twoLinesIntersectAtPoint L23 L31 Y ∧ Y.onCircle Ω ∧\n    twoLinesIntersectAtPoint L31 L12 Z ∧ Z.onCircle Ω := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2000_P6 :\n  ∀ (A B C H1 H2 H3 T1 T2 T3 X Y Z : Point)\n    (AB BC CA T12 T23 T31 L12 L23 L31 : Line) (Ω : Circle),\n    formAcuteTriangle A B C AB BC CA ∧\n    foot A H1 BC ∧\n    foot B H2 CA ∧\n    foot C H3 AB ∧\n    T1.onLine BC ∧ T1.onCircle Ω ∧\n    T2.onLine CA ∧ T2.onCircle Ω ∧\n    T3.onLine AB ∧ T3.onCircle Ω ∧\n    distinctPointsOnLine T1 T2 T12 ∧\n    distinctPointsOnLine T2 T3 T23 ∧\n    distinctPointsOnLine T3 T1 T31 ∧\n    symmetricLine H1H2 T12 L12 ∧\n    symmetricLine H2H3 T23 L23 ∧\n    symmetricLine H3H1 T31 L31 →\n    twoLinesIntersectAtPoint L12 L23 X ∧ X.onCircle Ω ∧\n    twoLinesIntersectAtPoint L23 L31 Y ∧ Y.onCircle Ω ∧\n    twoLinesIntersectAtPoint L31 L12 Z ∧ Z.onCircle Ω := by"}
+{"name": "IMO_2013_P4", "uuid": "b4068335-d3c6-4df8-89cf-0ccb0c3d918f", "statement_id": "af927829-8410-4b57-98d5-abe7f1505387", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2013_P4 :\n  ∀ (A B C H W M N X Y O1 O2 : Point) (AB BC CA : Line) (ω1 ω2 : Circle),\n    formAcuteTriangle A B C AB BC CA ∧\n    between B W C ���\n    foot B M CA ∧\n    foot C N AB ∧\n    coll B H M ∧\n    coll C H N ∧\n    circumCircle ω1 B W N ∧\n    diameter W X O1 ω1 ∧\n    circumCircle ω2 C W M ∧\n    diameter W Y O2 ω2 →\n    coll X H Y := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2013_P4 :\n  ∀ (A B C H W M N X Y O1 O2 : Point) (AB BC CA : Line) (ω1 ω2 : Circle),\n    formAcuteTriangle A B C AB BC CA ∧\n    between B W C ∧\n    foot B M CA ∧\n    foot C N AB ∧\n    coll B H M ∧\n    coll C H N ∧\n    circumCircle ω1 B W N ∧\n    diameter W X O1 ω1 ∧\n    circumCircle ω2 C W M ∧\n    diameter W Y O2 ω2 →\n    coll X H Y := by"}
+{"name": "IMO_2000_P1", "uuid": "d276789d-87e3-4ec9-86e2-27fb45d3b6e8", "statement_id": "e095e8b0-2d46-43b6-af22-fe89362f9d58", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2000_P1 :\n  ∀ (M N A B C D E P Q O1 O2 : Point) (G1 G2 : Circle) (AB CD AC BD AN BN : Line),\n    circlesIntersectsAtTwoPoints G1 G2 M N ∧\n    distinctPointsOnLine A B AB ∧\n    tangentAtPoint A O1 AB G1 ∧\n    tangentAtPoint B O2 AB G2 ∧\n    ¬ AB.intersectsLine CD ∧\n    distinctPointsOnLine M C CD ∧\n    C.onCircle G1 ∧ C ≠ M ∧\n    D.onCircle G2 ∧ D.onLine CD ∧ D ≠ M ∧\n    distinctPointsOnLine A C AC ∧\n    distinctPointsOnLine B D BD ∧\n    twoLinesIntersectAtPoint AC BD E ∧\n    distinctPointsOnLine A N AN ∧\n    twoLinesIntersectAtPoint AN CD P ∧\n    distinctPointsOnLine B N BN ∧\n    twoLinesIntersectAtPoint BN CD Q →\n    |(E─P)| = |(E─Q)| := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2000_P1 :\n  ∀ (M N A B C D E P Q O1 O2 : Point) (G1 G2 : Circle) (AB CD AC BD AN BN : Line),\n    circlesIntersectsAtTwoPoints G1 G2 M N ∧\n    distinctPointsOnLine A B AB ∧\n    tangentAtPoint A O1 AB G1 ∧\n    tangentAtPoint B O2 AB G2 ∧\n    ¬ AB.intersectsLine CD ∧\n    distinctPointsOnLine M C CD ∧\n    C.onCircle G1 ∧ C ≠ M ∧\n    D.onCircle G2 ∧ D.onLine CD ∧ D ≠ M ∧\n    distinctPointsOnLine A C AC ∧\n    distinctPointsOnLine B D BD ∧\n    twoLinesIntersectAtPoint AC BD E ∧\n    distinctPointsOnLine A N AN ∧\n    twoLinesIntersectAtPoint AN CD P ∧\n    distinctPointsOnLine B N BN ∧\n    twoLinesIntersectAtPoint BN CD Q →\n    |(E─P)| = |(E─Q)| := by"}
+{"name": "IMO_2010_P4", "uuid": "72a2ed90-0a4a-4002-8efd-cd69a46ae88a", "statement_id": "6c2cfe1f-27e9-4a5a-8a47-7b3988862191", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2010_P4 :\n  ∀ (A B C P K L M S O : Point) (t : Line) (Γ : Circle) (AB BC CA : Line),\n    formTriangle A B C AB BC CA ∧\n    |(C─A)| ≠ |(C─B)| ∧\n    insideTriangle P A B C AB BC CA ∧\n    O.isCentre Γ ∧\n    circumCircle Γ A B C ∧\n    between A P K ∧\n    between B P L ∧\n    between C P M ∧\n    K.onCircle Γ ∧\n    L.onCircle Γ ∧\n    M.onCircle Γ ∧\n    tangentAtPoint C O t Γ ∧\n    S.onLine AB ∧\n    S.onLine t ∧\n    |(S─C)| = |(S─P)| →\n    |(M─K)| = |(M─L)| := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2010_P4 :\n  ∀ (A B C P K L M S O : Point) (t : Line) (Γ : Circle) (AB BC CA : Line),\n    formTriangle A B C AB BC CA ∧\n    |(C─A)| ≠ |(C─B)| ∧\n    insideTriangle P A B C AB BC CA ∧\n    O.isCentre Γ ∧\n    circumCircle Γ A B C ∧\n    between A P K ∧\n    between B P L ∧\n    between C P M ∧\n    K.onCircle Γ ∧\n    L.onCircle Γ ∧\n    M.onCircle Γ ∧\n    tangentAtPoint C O t Γ ∧\n    S.onLine AB ∧\n    S.onLine t ∧\n    |(S─C)| = |(S─P)| →\n    |(M─K)| = |(M─L)| := by"}
+{"name": "IMO_2009_P4", "uuid": "78f6fe09-9a98-4152-9f17-f86f82761e40", "statement_id": "9b2d0823-07b8-4747-b656-503d413ea205", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2009_P4 :\n  ∀ (A B C D E K : Point) (AB BC CA AD DC : Line),\n    formTriangle A B C AB BC CA ∧ |(A─B)| = |(A─C)| ∧\n    between B D C ∧ ∠ C:A:D = ∠ D:A:B ∧\n    between C E A ∧ ∠ A:B:E = ∠ E:B:C ∧\n    formTriangle A D C AD DC CA ∧ inCentre K A D C ∧\n    ∠ B:E:K = ∟/2 →\n    (∠ C:A:B = ∟) ∨ (∠ C:A:B = 2/3 * ∟) := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2009_P4 :\n  ∀ (A B C D E K : Point) (AB BC CA AD DC : Line),\n    formTriangle A B C AB BC CA ∧ |(A─B)| = |(A─C)| ∧\n    between B D C ∧ ∠ C:A:D = ∠ D:A:B ∧\n    between C E A ∧ ∠ A:B:E = ∠ E:B:C ∧\n    formTriangle A D C AD DC CA ∧ inCentre K A D C ∧\n    ∠ B:E:K = ∟/2 →\n    (∠ C:A:B = ∟) ∨ (∠ C:A:B = 2/3 * ∟) := by"}
+{"name": "IMO_2001_P5", "uuid": "938ab8f5-ab6c-445b-8d22-12aefb878f0c", "statement_id": "96508c85-6efc-40cd-ad37-04ecc5014dcd", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2001_P5 :\n  ∀ (A B C P Q : Point) (AB BC CA AP BQ : Line),\n    formTriangle A B C AB BC CA ∧\n    ∠ B:A:C = 2/3 * ∟ ∧\n    distinctPointsOnLine A P AP ∧ P.onLine BC ∧ ∠ B:A:P = ∠ P:A:C ∧\n    distinctPointsOnLine B Q BQ ∧ Q.onLine CA ∧ ∠ A:B:Q = ∠ Q:B:C ∧\n    |(A─B)| + |(B─P)| = |(A─Q)| + |(Q─B)| →\n    ∠ A:B:C = 8/9 * ∟ ∧ ∠ B:C:A = 4/9 * ∟ := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2001_P5 :\n  ∀ (A B C P Q : Point) (AB BC CA AP BQ : Line),\n    formTriangle A B C AB BC CA ∧\n    ∠ B:A:C = 2/3 * ∟ ∧\n    distinctPointsOnLine A P AP ∧ P.onLine BC ∧ ∠ B:A:P = ∠ P:A:C ∧\n    distinctPointsOnLine B Q BQ ∧ Q.onLine CA ∧ ∠ A:B:Q = ∠ Q:B:C ∧\n    |(A─B)| + |(B─P)| = |(A─Q)| + |(Q─B)| →\n    ∠ A:B:C = 8/9 * ∟ ∧ ∠ B:C:A = 4/9 * ∟ := by"}
+{"name": "IMO_2014_P3", "uuid": "a4c01bdc-9fe3-4491-ae8e-ad59dcab51bc", "statement_id": "46affec7-ce09-4892-9659-a15d5880ecd5", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2014_P3 :\n  ∀ (A B C D H S T O: Point)\n    (AB BC CD DA BD SC CT ST: Line)\n    (Ω: Circle),\n    formQuadrilateral A B C D AB BC CD DA ∧\n    ∠ A:B:C = ∟ ∧\n    ∠ C:D:A = ∟ ∧\n    foot A H BD ∧\n    between A S B ∧\n    between A T D ∧\n    distinctPointsOnLine S C SC ∧\n    distinctPointsOnLine C T CT ∧\n    distinctPointsOnLine T S ST ∧\n    insideTriangle H S C T SC CT ST ∧\n    ∠ C:H:S = ∠ C:S:B + ∟ ∧\n    ∠ T:H:C = ∠ D:T:C + ∟ →\n    circumCircle Ω T S H ∧ circumCentre O T S H ∧ tangentAtPoint H O BD Ω := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem IMO_2014_P3 :\n  ∀ (A B C D H S T O: Point)\n    (AB BC CD DA BD SC CT ST: Line)\n    (Ω: Circle),\n    formQuadrilateral A B C D AB BC CD DA ∧\n    ∠ A:B:C = ∟ ∧\n    ∠ C:D:A = ∟ ∧\n    foot A H BD ∧\n    between A S B ∧\n    between A T D ∧\n    distinctPointsOnLine S C SC ∧\n    distinctPointsOnLine C T CT ∧\n    distinctPointsOnLine T S ST ∧\n    insideTriangle H S C T SC CT ST ∧\n    ∠ C:H:S = ∠ C:S:B + ∟ ∧\n    ∠ T:H:C = ∠ D:T:C + ∟ →\n    circumCircle Ω T S H ∧ circumCentre O T S H ∧ tangentAtPoint H O BD Ω := by"}
+{"name": "library7", "uuid": "3f61c5e7-2a86-4a6b-982e-c976149cd697", "statement_id": "1345d23e-349c-4602-acac-8c6c64fada47", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem library7 : ∀ (a b c d e : Point), triangle a b c ∧ midpoint a d b ∧ midpoint a e c → |(b─c)| = |(d─e)| * 2:= by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem library7 : ∀ (a b c d e : Point), triangle a b c ∧ midpoint a d b ∧ midpoint a e c → |(b─c)| = |(d─e)| * 2:= by"}
+{"name": "library3", "uuid": "442724e8-7bc3-45e6-8a85-ec9518347ef4", "statement_id": "c8f8686b-38d3-48f3-bebe-eda38e5a6a61", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem library3:\n  ∀ (A B C D : Point) (AB BC CD DA : Line),\n    formQuadrilateral A B C D AB BC CD DA ∧\n    ∠A:B:C = ∠C:D:A ∧\n    ∠B:C:D = ∠D:A:B →\n    parallelogram A B C D AB BC CD DA := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem library3:\n  ∀ (A B C D : Point) (AB BC CD DA : Line),\n    formQuadrilateral A B C D AB BC CD DA ∧\n    ∠A:B:C = ∠C:D:A ∧\n    ∠B:C:D = ∠D:A:B →\n    parallelogram A B C D AB BC CD DA := by"}
+{"name": "library1", "uuid": "dc9c3cc2-d161-4c77-883e-0b2c56fa4183", "statement_id": "b13532e2-191a-4d61-b492-4f22bb0e8ab5", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem library1 : ∀ (a b c d o: Point) (O : Circle), a.onCircle O ∧ b.onCircle O ∧ c.onCircle O ∧ d.onCircle O ∧  o.isCentre O ∧ |(a─b)| = |(c─d)| → ∠a:o:b =∠ c:o:d := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem library1 : ∀ (a b c d o: Point) (O : Circle), a.onCircle O ∧ b.onCircle O ∧ c.onCircle O ∧ d.onCircle O ∧  o.isCentre O ∧ |(a─b)| = |(c─d)| → ∠a:o:b =∠ c:o:d := by"}
+{"name": "library5", "uuid": "942f9e94-c5e2-4bb3-946b-119b9bb5db67", "statement_id": "d633f8a9-a6e5-4271-8154-adfebb0a6eb2", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem library5 :\n  ∀ (A B C O : Point) (AB BC CA: Line) (Ω : Circle), (formTriangle A B C AB BC CA) ∧ (insideTriangle O A B C AB BC CA) ∧ (A.onCircle Ω) ∧ (B.onCircle Ω) ∧ (C.onCircle Ω) ∧ (O.isCentre Ω)\n    → ∠ A:O:B = ∠ A:C:B + ∠ A:C:B := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem library5 :\n  ∀ (A B C O : Point) (AB BC CA: Line) (Ω : Circle), (formTriangle A B C AB BC CA) ∧ (insideTriangle O A B C AB BC CA) ∧ (A.onCircle Ω) ∧ (B.onCircle Ω) ∧ (C.onCircle Ω) ∧ (O.isCentre Ω)\n    → ∠ A:O:B = ∠ A:C:B + ∠ A:C:B := by"}
+{"name": "library10", "uuid": "b38eb37b-2990-4681-9da8-963e26192615", "statement_id": "968c6aa0-1648-4b54-a4df-0519d70421f1", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem library10 : ∀ (A B C D: Point), triangle A B C ∧ ∠D:A:B = ∠D:A:C ∧ coll B D C → |(A─C)| * |(B─D)| = |(A─B)| * |(C─D)|:= by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem library10 : ∀ (A B C D: Point), triangle A B C ∧ ∠D:A:B = ∠D:A:C ∧ coll B D C → |(A─C)| * |(B─D)| = |(A─B)| * |(C─D)|:= by"}
+{"name": "library4", "uuid": "af7ee43c-499d-4e3f-984b-4d069a121401", "statement_id": "c0fdd78a-0983-4105-baf5-4e27fd6ffcd0", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem library4: ∀ (A B C D E : Point) (Ω: Circle),\n  A.onCircle Ω ∧ B.onCircle Ω ∧ C.onCircle Ω ∧ D.onCircle Ω ∧\n  distinctFourPoints A B C D ∧\n  between A E B ∧ between C E D → |(A─E)| * |(E─B)| = |(C─E)| * |(E─D)|:= by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem library4: ∀ (A B C D E : Point) (Ω: Circle),\n  A.onCircle Ω ∧ B.onCircle Ω ∧ C.onCircle Ω ∧ D.onCircle Ω ∧\n  distinctFourPoints A B C D ∧\n  between A E B ∧ between C E D → |(A─E)| * |(E─B)| = |(C─E)| * |(E─D)|:= by"}
+{"name": "library2", "uuid": "f404f85a-c484-489c-a1a5-357baa5b27a3", "statement_id": "2231f15f-81e3-4cf7-831a-704de973c787", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem library2 : ∀ (A B C D: Point) (AB:Line) (Ω : Circle), distinctPointsOnLine A B AB ∧ A.onCircle Ω ∧ B.onCircle Ω ∧  C.onCircle Ω ∧ ∠B:C:A + ∠B:D:A = ∟ + ∟ ∧ C.opposingSides D AB → D.onCircle Ω:= by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem library2 : ∀ (A B C D: Point) (AB:Line) (Ω : Circle), distinctPointsOnLine A B AB ∧ A.onCircle Ω ∧ B.onCircle Ω ∧  C.onCircle Ω ∧ ∠B:C:A + ∠B:D:A = ∟ + ∟ ∧ C.opposingSides D AB → D.onCircle Ω:= by"}
+{"name": "library8", "uuid": "881dd01a-4644-4acc-92c2-9cffe6df6b49", "statement_id": "0fd1a00e-a626-4b9a-9df1-f9884866043f", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem library8 : ∀(A B C : Point), rightTriangle A B C → |(B─C)| > |(A─B)| ∧  |(B─C)| > |(A─C)| := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem library8 : ∀(A B C : Point), rightTriangle A B C → |(B─C)| > |(A─B)| ∧  |(B─C)| > |(A─C)| := by"}
+{"name": "librar9", "uuid": "601516d1-509c-4d02-a527-c2a7c71422ca", "statement_id": "07a3a318-5e58-4186-97e0-e0b37f6c3a63", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem librar9 :\n∀ (L M T : Line) (A B C D : Point),\n  twoLinesIntersectAtPoint L T A ∧\n  twoLinesIntersectAtPoint M T B ∧\n  C.onLine L ∧\n  D.onLine M ∧\n  C.sameSide D T ∧ A ≠ B\n  ∧ ∠ C:A:B + ∠ A:B:D = ∟ + ∟ → (¬ L.intersectsLine M) := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem librar9 :\n∀ (L M T : Line) (A B C D : Point),\n  twoLinesIntersectAtPoint L T A ∧\n  twoLinesIntersectAtPoint M T B ∧\n  C.onLine L ∧\n  D.onLine M ∧\n  C.sameSide D T ∧ A ≠ B\n  ∧ ∠ C:A:B + ∠ A:B:D = ∟ + ∟ → (¬ L.intersectsLine M) := by"}
+{"name": "generate_theorem_1", "uuid": "60bb5803-04f1-4c3f-91f5-8c9cd4066686", "statement_id": "6e2770e2-eb0d-4f27-a6a8-1df2cd1d7bec", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem generate_theorem_1:\n  ∀ (O A B C D M N : Point) (Ω : Circle) (L_AB L_CD : Line),\n    O.isCentre Ω ∧\n    A.onCircle Ω ∧ B.onCircle Ω ∧ C.onCircle Ω ∧ D.onCircle Ω ∧\n    distinctPointsOnLine A B L_AB ∧\n    distinctPointsOnLine C D L_CD ∧\n    |(A─B)| = |(C─D)| ∧\n    foot O M L_AB ∧\n    foot O N L_CD\n    → |(O─M)| = |(O─N)| := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem generate_theorem_1:\n  ∀ (O A B C D M N : Point) (Ω : Circle) (L_AB L_CD : Line),\n    O.isCentre Ω ∧\n    A.onCircle Ω ∧ B.onCircle Ω ∧ C.onCircle Ω ∧ D.onCircle Ω ∧\n    distinctPointsOnLine A B L_AB ∧\n    distinctPointsOnLine C D L_CD ∧\n    |(A─B)| = |(C─D)| ∧\n    foot O M L_AB ∧\n    foot O N L_CD\n    → |(O─M)| = |(O─N)| := by"}
+{"name": "cyclic_collinearity_from_exterior_angle", "uuid": "6392a559-a820-48e9-9fa5-331f566b7ddf", "statement_id": "0c7e0c36-aef5-49bc-b9ce-9ba8b629aff1", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem cyclic_collinearity_from_exterior_angle :\n  ∀ (A B C D E F : Point) (AB BC CD DA : Line) (Ω : Circle),\n    cyclicQuadrilateral A B C D AB BC CD DA Ω ∧\n    between A B E ∧\n    A.sameSide F CD ∧\n    ∠ E:B:C = ∠ F:D:C\n  → coll A D F := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem cyclic_collinearity_from_exterior_angle :\n  ∀ (A B C D E F : Point) (AB BC CD DA : Line) (Ω : Circle),\n    cyclicQuadrilateral A B C D AB BC CD DA Ω ∧\n    between A B E ∧\n    A.sameSide F CD ∧\n    ∠ E:B:C = ∠ F:D:C\n  → coll A D F := by"}
+{"name": "parallelogram_angle_bisector_creates_isoceles", "uuid": "c66b2a9b-6278-4a63-bca9-3ffd5123c753", "statement_id": "64cefebd-5433-4cdb-92c7-fcd7a18539d4", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem parallelogram_angle_bisector_creates_isoceles:\n  ∀ (A B C D E : Point) (AB BC CD DA AE : Line),\n    parallelogram A B C D AB BC CD DA ∧\n    distinctPointsOnLine A E AE ∧\n    E.onLine CD ∧ between C E D ∧\n    ∠D:A:E = ∠E:A:B\n    → |(D─E)| = |(A─D)| := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem parallelogram_angle_bisector_creates_isoceles:\n  ∀ (A B C D E : Point) (AB BC CD DA AE : Line),\n    parallelogram A B C D AB BC CD DA ∧\n    distinctPointsOnLine A E AE ∧\n    E.onLine CD ∧ between C E D ∧\n    ∠D:A:E = ∠E:A:B\n    → |(D─E)| = |(A─D)| := by"}
+{"name": "angle_altitude_circumradius_equal", "uuid": "d7ff25c4-8c79-41c8-896e-2ba5179e254d", "statement_id": "ce07b17c-675f-496f-b7ea-de5a370b2165", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem angle_altitude_circumradius_equal:\n  ∀ (A B C O H : Point) (Ω : Circle) (BC AB AC: Line),\n    acuteTriangle A B C ∧\n    formTriangle A B C AB BC AC ∧\n    circumCircle Ω A B C ∧\n    O.isCentre Ω ∧\n    -- This condition holds for acute triangles, but we state it to use the given theorem easily\n    O.sameSide C AB ∧\n    foot A H BC\n    → ∠ O:A:B = ∠ H:A:C := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem angle_altitude_circumradius_equal:\n  ∀ (A B C O H : Point) (Ω : Circle) (BC AB AC: Line),\n    acuteTriangle A B C ∧\n    formTriangle A B C AB BC AC ∧\n    circumCircle Ω A B C ∧\n    O.isCentre Ω ∧\n    -- This condition holds for acute triangles, but we state it to use the given theorem easily\n    O.sameSide C AB ∧\n    foot A H BC\n    → ∠ O:A:B = ∠ H:A:C := by"}
+{"name": "exercise_angle_bisector_lengths", "uuid": "7df1ad3d-30ec-447e-b431-c04a88872cd1", "statement_id": "1772f197-170a-4c0f-87e3-7726ce079d4b", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem exercise_angle_bisector_lengths :\n  ∀ (A B C D: Point),\n    triangle A B C ∧\n    ∠B:A:D = ∠C:A:D ∧\n    coll B D C ∧\n    between B D C ∧\n    |(A─C)| * 2 = |(A─B)| * 3\n    → |(B─C)| * 2 = |(B─D)| * 5 := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem exercise_angle_bisector_lengths :\n  ∀ (A B C D: Point),\n    triangle A B C ∧\n    ∠B:A:D = ∠C:A:D ∧\n    coll B D C ∧\n    between B D C ∧\n    |(A─C)| * 2 = |(A─B)| * 3\n    → |(B─C)| * 2 = |(B─D)| * 5 := by"}
+{"name": "tangent_parallel_to_chord_collinear", "uuid": "1f659569-68fb-4436-bfea-009ecf1328ff", "statement_id": "351b8349-ae60-4001-b578-b23ad9357946", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem tangent_parallel_to_chord_collinear:\n  ∀ (Ω : Circle) (L_AB L_T : Line) (O A B M T: Point),\n    O.isCentre Ω ∧\n    A.onCircle Ω ∧ B.onCircle Ω ∧ T.onCircle Ω ∧\n    A ≠ B ∧\n    midpoint A M B ∧\n    distinctPointsOnLine A B L_AB ∧\n    tangentAtPoint T O L_T Ω ∧\n    (¬ L_AB.intersectsLine L_T)\n    → coll O M T := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem tangent_parallel_to_chord_collinear:\n  ∀ (Ω : Circle) (L_AB L_T : Line) (O A B M T: Point),\n    O.isCentre Ω ∧\n    A.onCircle Ω ∧ B.onCircle Ω ∧ T.onCircle Ω ∧\n    A ≠ B ∧\n    midpoint A M B ∧\n    distinctPointsOnLine A B L_AB ∧\n    tangentAtPoint T O L_T Ω ∧\n    (¬ L_AB.intersectsLine L_T)\n    → coll O M T := by"}
+{"name": "parallelogram_transversal_property", "uuid": "abdb04d1-eeef-4a8e-b783-a646963988ed", "statement_id": "298d8ee1-fdde-4899-9845-9e32f11ec0c0", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem parallelogram_transversal_property:\n  ∀ (A B C D E F : Point) (AB AD BC CD CEFD : Line),\n    parallelogram A B C D AB AD BC CD ∧\n    threePointsOnLine C E F CEFD ∧\n    E.onLine AB ∧ F.onLine AD ∧ A ≠ E ∧ A ≠ F ∧\n    between A B E ∧ between A D F\n    → |(A─B)| * |(A─F)| + |(A─D)| * |(A─E)| = |(A─E)| * |(A─F)| := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem parallelogram_transversal_property:\n  ∀ (A B C D E F : Point) (AB AD BC CD CEFD : Line),\n    parallelogram A B C D AB AD BC CD ∧\n    threePointsOnLine C E F CEFD ∧\n    E.onLine AB ∧ F.onLine AD ∧ A ≠ E ∧ A ≠ F ∧\n    between A B E ∧ between A D F\n    → |(A─B)| * |(A─F)| + |(A─D)| * |(A─E)| = |(A─E)| * |(A─F)| := by"}
+{"name": "power_of_midpoint_on_median", "uuid": "63c3351d-0ae5-49bf-8e8a-f51328ab20cf", "statement_id": "d458e185-4c08-464f-9b47-eda793dbaca0", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem power_of_midpoint_on_median:\n  ∀ (A B C M D O : Point) (Ω : Circle),\n    (triangle A B C) ∧\n    (midpoint B M C) ∧\n    (circumCircle Ω A B C) ∧\n    (O.isCentre Ω) ∧\n    (D.onCircle Ω) ∧\n    (D ≠ A) ∧\n    (between A M D)\n    → |(A─M)| * |(M─D)| = |(B─M)| * |(B─M)| := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem power_of_midpoint_on_median:\n  ∀ (A B C M D O : Point) (Ω : Circle),\n    (triangle A B C) ∧\n    (midpoint B M C) ∧\n    (circumCircle Ω A B C) ∧\n    (O.isCentre Ω) ∧\n    (D.onCircle Ω) ∧\n    (D ≠ A) ∧\n    (between A M D)\n    → |(A─M)| * |(M─D)| = |(B─M)| * |(B─M)| := by"}
+{"name": "orthocenter_power_property", "uuid": "dbb4b068-9b10-4ec3-8172-f37553052c63", "statement_id": "c9886d90-ece9-42f3-9ee8-3fe4068662d3", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem orthocenter_power_property:\n  ∀ (A B C D E F H : Point) (AB BC CA : Line),\n    acuteTriangle A B C ∧\n    orthoCentre H A B C D E F AB BC CA\n    → |(A─H)| * |(H─D)| = |(B─H)| * |(H─E)| := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem orthocenter_power_property:\n  ∀ (A B C D E F H : Point) (AB BC CA : Line),\n    acuteTriangle A B C ∧\n    orthoCentre H A B C D E F AB BC CA\n    → |(A─H)| * |(H─D)| = |(B─H)| * |(H─E)| := by"}
+{"name": "iso_triangle_median_property", "uuid": "aa14dbcc-1b53-4d02-974a-5d785b2df56d", "statement_id": "e1180dc3-0ba9-4d4e-85ed-c5f9b630201d", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem iso_triangle_median_property:\n  ∀ (A B C M P : Point),\n    isoTriangle A B C ∧ midpoint B M C ∧ coll A M P\n    → |(P─B)| = |(P─C)| := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem iso_triangle_median_property:\n  ∀ (A B C M P : Point),\n    isoTriangle A B C ∧ midpoint B M C ∧ coll A M P\n    → |(P─B)| = |(P─C)| := by"}
+{"name": "right_triangles_share_hypotenuse_iso_angles", "uuid": "caeb8d4b-efbd-4b11-ba69-78fe21297048", "statement_id": "8df7ab97-3334-420c-a247-fdd8b8cd253b", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem right_triangles_share_hypotenuse_iso_angles:\n  ∀ (A B C D M : Point),\n    rightTriangle B A C ∧\n    rightTriangle D A C ∧\n    midpoint A M C ∧\n    triangle M B D\n    → ∠ M:B:D = ∠ M:D:B := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem right_triangles_share_hypotenuse_iso_angles:\n  ∀ (A B C D M : Point),\n    rightTriangle B A C ∧\n    rightTriangle D A C ∧\n    midpoint A M C ∧\n    triangle M B D\n    → ∠ M:B:D = ∠ M:D:B := by"}
+{"name": "incenter_of_isosceles_triangle_is_isosceles", "uuid": "808342b1-95ab-497b-a1ca-e20a14b1f904", "statement_id": "7b09bc49-d422-489c-833b-d8f9b1e98ae4", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem incenter_of_isosceles_triangle_is_isosceles :\n  ∀ (A B C I : Point),\n    isoTriangle A B C ∧\n    inCentre I A B C\n    → isoTriangle I B C := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem incenter_of_isosceles_triangle_is_isosceles :\n  ∀ (A B C I : Point),\n    isoTriangle A B C ∧\n    inCentre I A B C\n    → isoTriangle I B C := by"}
+{"name": "Numina_Geometry_1110", "uuid": "8743b3d1-10ab-46da-9de4-7e6c341c7a65", "statement_id": "86a532ab-0f0d-4b11-861a-724c9eb0e1d9", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Numina_Geometry_1110 :\n  ∀ (A B C H M K : Point) (AC: Line),\n    (triangle A B C) ∧\n    (between A H C) ∧\n    (foot B H AC) ∧\n    (distinctPointsOnLine A C AC) ∧\n    (midpoint B M C) ∧\n    (midpoint A K B)\n    →\n    (∠ K:H:M = ∠ A:B:C) ∧\n    (∠ K:M:H = ∠ A:C:B) ∧\n    (∠ H:K:M = ∠ B:A:C) := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Numina_Geometry_1110 :\n  ∀ (A B C H M K : Point) (AC: Line),\n    (triangle A B C) ∧\n    (between A H C) ∧\n    (foot B H AC) ∧\n    (distinctPointsOnLine A C AC) ∧\n    (midpoint B M C) ∧\n    (midpoint A K B)\n    →\n    (∠ K:H:M = ∠ A:B:C) ∧\n    (∠ K:M:H = ∠ A:C:B) ∧\n    (∠ H:K:M = ∠ B:A:C) := by"}
+{"name": "right_triangle_altitude_theorem", "uuid": "cd7daa78-6db8-4b36-a01c-027145f6dfd8", "statement_id": "6626cb09-df11-4ea1-8dc6-ee4ab61ab5ea", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem right_triangle_altitude_theorem:\n  ∀ (A B C D: Point) (BC : Line),\n    rightTriangle A B C ∧\n    distinctPointsOnLine B C BC ∧\n    foot A D BC\n    → |(A─D)| * |(A─D)| = |(B─D)| * |(D─C)| := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem right_triangle_altitude_theorem:\n  ∀ (A B C D: Point) (BC : Line),\n    rightTriangle A B C ∧\n    distinctPointsOnLine B C BC ∧\n    foot A D BC\n    → |(A─D)| * |(A─D)| = |(B─D)| * |(D─C)| := by"}
+{"name": "Numina_Geometry_599", "uuid": "5ff16ac6-00a0-4d15-aa78-0def5d35894a", "statement_id": "b09f222f-7574-46fa-917b-771baa862b8e", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Numina_Geometry_599 :\n  ∀ (A B C D E F : Point) (AB BC CD DA : Line),\n    rectangle A B C D AB BC CD DA ∧\n    midpoint B E C ∧\n    midpoint C F D →\n    |(A─E)| < |(E─F)| + |(E─F)| := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Numina_Geometry_599 :\n  ∀ (A B C D E F : Point) (AB BC CD DA : Line),\n    rectangle A B C D AB BC CD DA ∧\n    midpoint B E C ∧\n    midpoint C F D →\n    |(A─E)| < |(E─F)| + |(E─F)| := by"}
+{"name": "right_triangle_median_perpendicular_midpoint", "uuid": "5cbc2d9d-b321-4fec-857c-30acb150c8ce", "statement_id": "ef1de981-c037-40d8-a1f9-5cbf6b252273", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem right_triangle_median_perpendicular_midpoint:\n  ∀ (A B C M D : Point) (AB BC : Line),\n    rightTriangle A B C ∧\n    distinctPointsOnLine A B AB ∧\n    distinctPointsOnLine B C BC ∧\n    midpoint B M C ∧\n    foot M D AB\n    → midpoint A D B := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem right_triangle_median_perpendicular_midpoint:\n  ∀ (A B C M D : Point) (AB BC : Line),\n    rightTriangle A B C ∧\n    distinctPointsOnLine A B AB ∧\n    distinctPointsOnLine B C BC ∧\n    midpoint B M C ∧\n    foot M D AB\n    → midpoint A D B := by"}
+{"name": "Apollonius_Theorem", "uuid": "e9e690ee-a7e1-4cd9-9369-fc0112a314f2", "statement_id": "cf6b45d2-7732-41b9-ac29-9bf7b66a034c", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Apollonius_Theorem :\n  ∀ (A B C D : Point),\n    triangle A B C ∧\n    midpoint B D C\n    → |(A─B)| * |(A─B)| + |(A─C)| * |(A─C)| = 2 * (|(A─D)| * |(A─D)| + |(B─D)| * |(B─D)|) := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Apollonius_Theorem :\n  ∀ (A B C D : Point),\n    triangle A B C ∧\n    midpoint B D C\n    → |(A─B)| * |(A─B)| + |(A─C)| * |(A─C)| = 2 * (|(A─D)| * |(A─D)| + |(B─D)| * |(B─D)|) := by"}
+{"name": "Numina_Geometry_808", "uuid": "732dc8c9-c5df-42b5-8202-43ca00d95449", "statement_id": "92180038-daa9-4177-853b-afc4da860cb7", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Numina_Geometry_808 :\n  ∀ (A B C D : Point) (AB BC CD DA : Line),\n    (formQuadrilateral A B C D AB BC CD DA) ∧\n    (∠C:B:D = ∠C:A:B) ∧\n    (∠A:C:D = ∠B:D:A) →\n    ∠A:B:C = ∠A:D:C := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Numina_Geometry_808 :\n  ∀ (A B C D : Point) (AB BC CD DA : Line),\n    (formQuadrilateral A B C D AB BC CD DA) ∧\n    (∠C:B:D = ∠C:A:B) ∧\n    (∠A:C:D = ∠B:D:A) →\n    ∠A:B:C = ∠A:D:C := by"}
+{"name": "Numina_Geometry_623", "uuid": "76115943-9486-4f8c-b89d-26f5c86fb688", "statement_id": "9d92ddef-cbf5-400e-a419-c17ce97e1f19", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Numina_Geometry_623 :\n  ∀ (A B C K L M : Point),\n    rightTriangle C A B ∧\n    isoTriangle C A B ∧\n    midpoint A K B ∧\n    between B L C ∧\n    between A M C ∧\n    |(B─L)| = |(C─M)| →\n    rightTriangle K L M ∧\n    isoTriangle K L M := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Numina_Geometry_623 :\n  ∀ (A B C K L M : Point),\n    rightTriangle C A B ∧\n    isoTriangle C A B ∧\n    midpoint A K B ∧\n    between B L C ∧\n    between A M C ∧\n    |(B─L)| = |(C─M)| →\n    rightTriangle K L M ∧\n    isoTriangle K L M := by"}
+{"name": "median_is_half_side_implies_right_triangle", "uuid": "b56607ff-4cc4-4d20-954d-5659e3fdc6f4", "statement_id": "3ecdcb05-3c33-4d3c-8149-e9c70afd584f", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem median_is_half_side_implies_right_triangle:\n  ∀ (A B C M : Point),\n    triangle A B C ∧\n    midpoint B M C ∧\n    |(A─M)| = |(B─M)|\n    → ∠ B:A:C = ∟ := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem median_is_half_side_implies_right_triangle:\n  ∀ (A B C M : Point),\n    triangle A B C ∧\n    midpoint B M C ∧\n    |(A─M)| = |(B─M)|\n    → ∠ B:A:C = ∟ := by"}
+{"name": "Numina_Geometry_800", "uuid": "509f3ad0-6c11-41d4-ae4c-040edd952cfb", "statement_id": "188f9826-79a6-43f1-bee9-58d50e52c9d3", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Numina_Geometry_800 : ∀ (A B C L X Y : Point) (XL BC: Line) (Ω : Circle),\n  triangle A B C ∧\n  between A L C ∧\n  between A X B ∧\n  (∠ A:L:B = ∠ B:L:C) ∧\n  distinctPointsOnLine X L XL ∧\n  distinctPointsOnLine B C BC ∧\n  (¬ XL.intersectsLine BC) ∧\n  circumCircle Ω A X C ∧\n  Y.onCircle Ω ∧\n  between B Y C\n  → |(A─X)| = |(B─Y)| := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Numina_Geometry_800 : ∀ (A B C L X Y : Point) (XL BC: Line) (Ω : Circle),\n  triangle A B C ∧\n  between A L C ∧\n  between A X B ∧\n  (∠ A:L:B = ∠ B:L:C) ∧\n  distinctPointsOnLine X L XL ∧\n  distinctPointsOnLine B C BC ∧\n  (¬ XL.intersectsLine BC) ∧\n  circumCircle Ω A X C ∧\n  Y.onCircle Ω ∧\n  between B Y C\n  → |(A─X)| = |(B─Y)| := by"}
+{"name": "Numina_Geometry_1011", "uuid": "1ad804c5-fd92-46dc-86ab-6d1340a4e162", "statement_id": "ec443596-205d-4d5e-bd0e-d9688e5bc153", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Numina_Geometry_1011 :\n  ∀ (A B C M N : Point),\n    (triangle A B C) ∧\n    (|(A─C)| ≥ |(B─C)| ∧ |(A─C)| ≥ |(A─B)|) ∧\n    (between A M C) ∧\n    (between A N C) ∧\n    (|(A─M)| = |(A─B)|) ∧\n    (|(C─N)| = |(C─B)|) ∧\n    (|(B─M)| = |(B─N)|) →\n    isoTriangle B A C := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Numina_Geometry_1011 :\n  ∀ (A B C M N : Point),\n    (triangle A B C) ∧\n    (|(A─C)| ≥ |(B─C)| ∧ |(A─C)| ≥ |(A─B)|) ∧\n    (between A M C) ∧\n    (between A N C) ∧\n    (|(A─M)| = |(A─B)|) ∧\n    (|(C─N)| = |(C─B)|) ∧\n    (|(B─M)| = |(B─N)|) →\n    isoTriangle B A C := by"}
+{"name": "Numina_Geometry_557", "uuid": "0620373c-ef87-4b22-b384-cddedcbfb8b7", "statement_id": "c66ed48c-f9eb-4c2d-8e0e-85640e05b22f", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Numina_Geometry_557 :\n  ∀ (a b c d e : Point),\n    (triangle a b c) ∧\n    coll b c d ∧\n    coll a c e ∧\n    ∠ a:e:b = ∟ ∧\n    ∠ a:d:b = ∟ ∧ e ≠ a ∧ d ≠ b ∧\n    (|(a─d)| = |(b─e)|)\n    → isoTriangle c a b := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Numina_Geometry_557 :\n  ∀ (a b c d e : Point),\n    (triangle a b c) ∧\n    coll b c d ∧\n    coll a c e ∧\n    ∠ a:e:b = ∟ ∧\n    ∠ a:d:b = ∟ ∧ e ≠ a ∧ d ≠ b ∧\n    (|(a─d)| = |(b─e)|)\n    → isoTriangle c a b := by"}
+{"name": "Numina_Geometry_856", "uuid": "e5b8cf22-e83c-4576-b4e0-ed1393324b3c", "statement_id": "8ef8fd3c-f04b-4313-a93e-8d268c7b4f8a", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Numina_Geometry_856 :\n  ∀ (A B C D E F P : Point) (AB BC CD DA : Line),\n    (parallelogram A B C D AB BC CD DA) ∧\n    |(A─B)| = |(B─C)| ∧\n    |(B─C)| = |(C─D)| ∧\n    |(C─D)| = |(D─A)| ∧\n    midpoint A E B ∧\n    midpoint B F C ∧\n    |(P─A)| = |(P─F)| ∧\n    |(P─E)| = |(P─C)|\n    → coll B D P := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Numina_Geometry_856 :\n  ∀ (A B C D E F P : Point) (AB BC CD DA : Line),\n    (parallelogram A B C D AB BC CD DA) ∧\n    |(A─B)| = |(B─C)| ∧\n    |(B─C)| = |(C─D)| ∧\n    |(C─D)| = |(D─A)| ∧\n    midpoint A E B ∧\n    midpoint B F C ∧\n    |(P─A)| = |(P─F)| ∧\n    |(P─E)| = |(P─C)|\n    → coll B D P := by"}
+{"name": "Numina_Geometry_1652", "uuid": "1fa576c6-6c84-4198-8f7a-a0d44a1f9539", "statement_id": "08bab781-11b1-4f8f-b25f-2e9070e4c6d6", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Numina_Geometry_1652 :\n  ∀ (A B C D E G : Point) (AD BE : Line),\n    triangle A B C ∧\n    midpoint B D C ∧\n    midpoint A E C ∧\n    distinctPointsOnLine A D AD ∧\n    distinctPointsOnLine B E BE ∧\n    twoLinesIntersectAtPoint AD BE G ∧\n    |(A─G)| = |(B─G)| ∧\n    |(B─G)| = |(C─G)|\n    → |(A─B)| = |(B─C)| ∧ |(B─C)| = |(C─A)| := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Numina_Geometry_1652 :\n  ∀ (A B C D E G : Point) (AD BE : Line),\n    triangle A B C ∧\n    midpoint B D C ∧\n    midpoint A E C ∧\n    distinctPointsOnLine A D AD ∧\n    distinctPointsOnLine B E BE ∧\n    twoLinesIntersectAtPoint AD BE G ∧\n    |(A─G)| = |(B─G)| ∧\n    |(B─G)| = |(C─G)|\n    → |(A─B)| = |(B─C)| ∧ |(B─C)| = |(C─A)| := by"}
+{"name": "Numina_Geometry_850", "uuid": "8fc684ae-e19c-44c1-b8ba-dbef5c08b847", "statement_id": "8306c4a6-a47b-46d4-a26d-90bd4b220991", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Numina_Geometry_850 : ∀ (A B C I O : Point) (Ω : Circle), (triangle A B C) ∧ (inCentre I A B C) ∧ (circumCentre O A I C) ∧ (circumCircle Ω A B C) → O.onCircle Ω := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Numina_Geometry_850 : ∀ (A B C I O : Point) (Ω : Circle), (triangle A B C) ∧ (inCentre I A B C) ∧ (circumCentre O A I C) ∧ (circumCircle Ω A B C) → O.onCircle Ω := by"}
+{"name": "Numina_Geometry_1200", "uuid": "67c4ff6b-4856-46b3-a349-a96f33db1202", "statement_id": "b4377270-3cc5-4715-8edf-3a503a8779f0", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Numina_Geometry_1200 :\n  ∀ (A B C K E : Point),\n  formTriangle A B C AB BC CA ∧\n  midpoint B K C ∧\n  foot A E BC ∧\n  ∠ A:B:C = ∠ A:C:B + ∠ A:C:B\n    → |(A─B)| = |(E─K)| + |(E─K)| := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Numina_Geometry_1200 :\n  ∀ (A B C K E : Point),\n  formTriangle A B C AB BC CA ∧\n  midpoint B K C ∧\n  foot A E BC ∧\n  ∠ A:B:C = ∠ A:C:B + ∠ A:C:B\n    → |(A─B)| = |(E─K)| + |(E─K)| := by"}
+{"name": "Numina_Geometry_1633", "uuid": "68205ae6-a4d9-45b7-8e56-3c0797f0726f", "statement_id": "a808c125-28d2-4567-9ab9-99b84fd0faab", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Numina_Geometry_1633 :\n  ∀ (A B C M N : Point),\n    (triangle A B C) ∧\n    (between A M B) ∧\n    (between B N C) ∧\n    (∠M:N:B = ∠A:C:B) ∧\n    (|(B─N)| = 1) ∧\n    (|(M─N)| = 2) ∧\n    (|(A─M)| = 3) →\n    |(A─C)| > 4 := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Numina_Geometry_1633 :\n  ∀ (A B C M N : Point),\n    (triangle A B C) ∧\n    (between A M B) ∧\n    (between B N C) ∧\n    (∠M:N:B = ∠A:C:B) ∧\n    (|(B─N)| = 1) ∧\n    (|(M─N)| = 2) ∧\n    (|(A─M)| = 3) →\n    |(A─C)| > 4 := by"}
+{"name": "Numina_Geometry_705", "uuid": "0d87e1c7-5ca0-4ffe-9b0b-2d2f5178582b", "statement_id": "c0d25c2d-7cfa-4aae-acdc-480bc5a4c443", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Numina_Geometry_705 :\n  ∀ (S M Q T O : Point),\n    triangle S Q T ∧\n    coll Q T M ∧\n    (∠ Q:S:M = ∠ M:S:T) ∧\n    between S O T ∧\n    (∠ O:Q:T = ∠ Q:T:S + ∠ Q:S:T)\n    → ∠ Q:O:M = ∠ M:O:T := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Numina_Geometry_705 :\n  ∀ (S M Q T O : Point),\n    triangle S Q T ∧\n    coll Q T M ∧\n    (∠ Q:S:M = ∠ M:S:T) ∧\n    between S O T ∧\n    (∠ O:Q:T = ∠ Q:T:S + ∠ Q:S:T)\n    → ∠ Q:O:M = ∠ M:O:T := by"}
+{"name": "Numina_Geometry_542", "uuid": "e41d3129-889d-4fc8-8f59-dac5997dd965", "statement_id": "ac2aa63e-acab-4207-bb3f-64cbccd1532e", "system": "You are an expert in mathematics and proving theorems in Lean 4.", "instruction": "Think about and solve the following problems step by step in Lean 4.\n# Formal Statement:\n\n```lean4\nimport Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Numina_Geometry_542 : ∀ (A B C H P Q : Point) (AB BC CA : Line),\n    (formAcuteTriangle A B C AB BC CA) ∧\n    foot A H BC ∧\n    foot H P AB ∧\n    foot H Q AC\n    → ∠B:Q:H = ∠C:P:H := by\n```\n", "natural_language": "", "formal_statement": "import Mathlib\nimport SystemE\nimport LeanGeo\nnamespace LeanGeo\n\ntheorem Numina_Geometry_542 : ∀ (A B C H P Q : Point) (AB BC CA : Line),\n    (formAcuteTriangle A B C AB BC CA) ∧\n    foot A H BC ∧\n    foot H P AB ∧\n    foot H Q AC\n    → ∠B:Q:H = ∠C:P:H := by"}