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{"name": "imo_2020_p1", "split": "test", "informal_prefix": "/--Consider the convex quadrilateral $ABCD$. The point $P$ is in the interior of $ABCD$. The following ratio equalities hold: $\\angle PAD : \\angle PBA : \\angle DPA = 1 : 2 : 3 = \\angle CBP : \\angle BAP : \\angle BPC$. Prove that the following three lines meet in a point: the internal bisectors of angles $\\angle ADP$ and $\\angle PCB$ and the perpendicular bisector of segment $\\overline{AB}$.-/\n", "formal_statement": "import SystemE\nimport UniGeo.Relations\nimport UniGeo.Abbre\nimport UniGeo.Theorem\n\n/--Consider the convex quadrilateral $ABCD$. The point $P$ is in the interior of $ABCD$. The following ratio equalities hold: $\\angle PAD : \\angle PBA : \\angle DPA = 1 : 2 : 3 = \\angle CBP : \\angle BAP : \\angle BPC$. Prove that the following three lines meet in a point: the internal bisectors of angles $\\angle ADP$ and $\\angle PCB$ and the perpendicular bisector of segment $\\overline{AB}$.-/\ntheorem problem_imo_2020_p1:\n ∀ (A B C D P Q R Z X : Point) (AB BC CD DA DQ l CR: Line),\n (formQuadrilateral A B C D AB BC CD DA) ∧\n (P.sameSide C AB) ∧ (P.sameSide A CD) ∧ (P.sameSide D BC) ∧\n (P.sameSide B DA) ∧ (B ≠ P) ∧ (A ≠ P) ∧ (C ≠ P) ∧ (D ≠ P) ∧\n (2 * (∠P : A : D : ℝ) = (∠P : B : A : ℝ)) ∧\n (3 * (∠P : A : D : ℝ) = (∠D : P : A : ℝ)) ∧\n (2 * (∠C : B : P : ℝ) = (∠C : A : P : ℝ)) ∧\n (3 * (∠C : B : P : ℝ) = (∠B : P : C : ℝ)) ∧\n (∠Q : D : A : ℝ) = (∠Q : D : P : ℝ) ∧\n (∠R : C : B : ℝ) = (∠R : C : B : ℝ) ∧\n (perpBisector A B l) ∧\n (distinctPointsOnLine D Q DQ) ∧\n (distinctPointsOnLine C R CR) →\n concurrent DQ CR l := by\n ", "goal": " ∀ (A B C D P Q R Z X : Point) (AB BC CD DA DQ l CR: Line),\n (formQuadrilateral A B C D AB BC CD DA) ∧\n (P.sameSide C AB) ∧ (P.sameSide A CD) ∧ (P.sameSide D BC) ∧\n (P.sameSide B DA) ∧ (B ≠ P) ∧ (A ≠ P) ∧ (C ≠ P) ∧ (D ≠ P) ∧\n (2 * (∠P : A : D : ℝ) = (∠P : B : A : ℝ)) ∧\n (3 * (∠P : A : D : ℝ) = (∠D : P : A : ℝ)) ∧\n (2 * (∠C : B : P : ℝ) = (∠C : A : P : ℝ)) ∧\n (3 * (∠C : B : P : ℝ) = (∠B : P : C : ℝ)) ∧\n (∠Q : D : A : ℝ) = (∠Q : D : P : ℝ) ∧\n (∠R : C : B : ℝ) = (∠R : C : B : ℝ) ∧\n (perpBisector A B l) ∧\n (distinctPointsOnLine D Q DQ) ∧\n (distinctPointsOnLine C R CR) →\n concurrent DQ CR l ", "header": "import SystemE\nimport UniGeo.Relations\nimport UniGeo.Abbre\nimport UniGeo.Theorem\n\n", "formal_proof": "", "uuid": "0f017945-400c-4108-afd6-018ed0cbb722", "statementid": "0f017945-400c-4108-afd6-018ed0cbb722", "natural_language": "Consider the convex quadrilateral $ABCD$. The point $P$ is in the interior of $ABCD$. The following ratio equalities hold: $\\angle PAD : \\angle PBA : \\angle DPA = 1 : 2 : 3 = \\angle CBP : \\angle BAP : \\angle BPC$. Prove that the following three lines meet in a point: the internal bisectors of angles $\\angle ADP$ and $\\angle PCB$ and the perpendicular bisector of segment $\\overline{AB}$.", "data_name": "Geometry", "category": "Geometry"}
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{"name": "imo_2021_p3", "split": "test", "informal_prefix": "/--Let $D$ be an interior point of the acute triangle $ABC$ with $AB > AC$ so that $\\angle DAB= \\angle CAD$. The point $E$ on the segment $AC$ satisfies $\\angle ADE= \\angle BCD$, the point $F$ on the segment $AB$ satisfies $\\angle FDA= \\angle DBC$, and the point $X$ on the line $AC$ satisfies $CX=BX$. Let $O_1$ and $O_2$ be the circumcentres of the triangles $ADC$ and $EXD$ respectively. Prove that the lines $BC$, $EF$, and $O_1 O_2$ are concurrent.-/\n", "formal_statement": "import SystemE\nimport UniGeo.Relations\nimport UniGeo.Abbre\nimport UniGeo.Theorem\n\n/--Let $D$ be an interior point of the acute triangle $ABC$ with $AB > AC$ so that $\\angle DAB= \\angle CAD$. The point $E$ on the segment $AC$ satisfies $\\angle ADE= \\angle BCD$, the point $F$ on the segment $AB$ satisfies $\\angle FDA= \\angle DBC$, and the point $X$ on the line $AC$ satisfies $CX=BX$. Let $O_1$ and $O_2$ be the circumcentres of the triangles $ADC$ and $EXD$ respectively. Prove that the lines $BC$, $EF$, and $O_1 O_2$ are concurrent.-/\ntheorem problem_imo_2021_p3: ∀ (A B C D E F X O₁ O₂: Point) (AB BC CA: Line), formAcuteTriangle A B C AB BC CA ∧\nD.sameSide A BC ∧\nD.sameSide C AB ∧\nD.sameSide B CA ∧\n|(A─B)| > |(A─C)| ∧\nE.onLine AC ∧ ∠A:D:E = ∠B:C:D ∧\nF.onLine AB ∧ ∠F:D:A = ∠D:B:C ∧\nX.onLine AC ∧ |(C─X)| = |(B─X)| ∧\ncircumCentre O₁ A D C ∧ circumCentre O₂ E X D → concurrent BC EF O₁O₂ := by\n ", "goal": " (A B C D E F X O₁ O₂: Point) (AB BC CA: Line), formAcuteTriangle A B C AB BC CA ∧\nD.sameSide A BC ∧\nD.sameSide C AB ∧\nD.sameSide B CA ∧\n|(A─B)| > |(A─C)| ∧\nE.onLine AC ∧ ∠A:D:E = ∠B:C:D ∧\nF.onLine AB ∧ ∠F:D:A = ∠D:B:C ∧\nX.onLine AC ∧ |(C─X)| = |(B─X)| ∧\ncircumCentre O₁ A D C ∧ circumCentre O₂ E X D → concurrent BC EF O₁O₂ ", "header": "import SystemE\nimport UniGeo.Relations\nimport UniGeo.Abbre\nimport UniGeo.Theorem\n\n", "formal_proof": "", "uuid": "56c84b0c-f8df-4024-b398-93b3596fbeda", "statementid": "56c84b0c-f8df-4024-b398-93b3596fbeda", "natural_language": "Let $D$ be an interior point of the acute triangle $ABC$ with $AB > AC$ so that $\\angle DAB= \\angle CAD$. The point $E$ on the segment $AC$ satisfies $\\angle ADE= \\angle BCD$, the point $F$ on the segment $AB$ satisfies $\\angle FDA= \\angle DBC$, and the point $X$ on the line $AC$ satisfies $CX=BX$. Let $O_1$ and $O_2$ be the circumcentres of the triangles $ADC$ and $EXD$ respectively. Prove that the lines $BC$, $EF$, and $O_1 O_2$ are concurrent.", "data_name": "Geometry", "category": "Geometry"}
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{"name": "imo_2021_p4", "split": "test", "informal_prefix": "/--Let $\\Gamma$ be a circle with centre $I$, and $ABCD$ a convex quadrilateral such that each of the segments $AB, BC, CD$ and $DA$ is tangent to $\\Gamma$. Let $\\Omega$ be the circumcircle of the triangle $AIC$. The extension of $BA$ beyond $A$ meets $\\Omega$ at $X$, and the extension of $BC$ beyond $C$ meets $\\Omega$ at $Z$. The extensions of $AD$ and $CD$ beyond $D$ meet $\\Omega$ at $Y$ and $T$, respectively. Prove that \\AD + DT + T X + XA = CD + DY + Y Z + ZC\\-/\n", "formal_statement": "import SystemE\nimport UniGeo.Relations\nimport UniGeo.Abbre\nimport UniGeo.Theorem\n\n/--Let $\\Gamma$ be a circle with centre $I$, and $ABCD$ a convex quadrilateral such that each of the segments $AB, BC, CD$ and $DA$ is tangent to $\\Gamma$. Let $\\Omega$ be the circumcircle of the triangle $AIC$. The extension of $BA$ beyond $A$ meets $\\Omega$ at $X$, and the extension of $BC$ beyond $C$ meets $\\Omega$ at $Z$. The extensions of $AD$ and $CD$ beyond $D$ meet $\\Omega$ at $Y$ and $T$, respectively. Prove that \\AD + DT + T X + XA = CD + DY + Y Z + ZC\\-/\ntheorem problem_imo_2021_p4 : ∀ (A B C D Y T Z X : Point) (AB BC CD DA: Line) (Γ Ω: Circle), formQuadrilateral A B C D AB BC CD DA ∧\ntangentLine AB Γ ∧ tangentLine BC Γ ∧ tangentLine CD Γ ∧ tangentLine DA Γ ∧\ncircumCircle Ω A I C ∧\nX.onLine AB ∧ X.onCircle Ω ∧\nZ.onLine BC ∧ Z.onCircle Ω ∧\nY.onLine DA ∧ Y.onCircle Ω ∧\nT.onLine CD ∧ T.onCircle Ω → |(A─D)| + |(D─T)| + |(T─X)| + |(X─A)| = |(C─D)| + |(D─Y)| + |(Y─Z)| + |(Z─C)|:= by\n ", "goal": "∀ (A B C D Y T Z X : Point) (AB BC CD DA: Line) (Γ Ω: Circle), formQuadrilateral A B C D AB BC CD DA ∧\ntangentLine AB Γ ∧ tangentLine BC Γ ∧ tangentLine CD Γ ∧ tangentLine DA Γ ∧\ncircumCircle Ω A I C ∧\nX.onLine AB ∧ X.onCircle Ω ∧\nZ.onLine BC ∧ Z.onCircle Ω ∧\nY.onLine DA ∧ Y.onCircle Ω ∧\nT.onLine CD ∧ T.onCircle Ω → |(A─D)| + |(D─T)| + |(T─X)| + |(X─A)| = |(C─D)| + |(D─Y)| + |(Y─Z)| + |(Z─C)|", "header": "import SystemE\nimport UniGeo.Relations\nimport UniGeo.Abbre\nimport UniGeo.Theorem\n\n", "formal_proof": "", "uuid": "99c6bcb9-c06e-41ce-89b8-28e3d873421b", "statementid": "99c6bcb9-c06e-41ce-89b8-28e3d873421b", "natural_language": "Let $\\Gamma$ be a circle with centre $I$, and $ABCD$ a convex quadrilateral such that each of the segments $AB, BC, CD$ and $DA$ is tangent to $\\Gamma$. Let $\\Omega$ be the circumcircle of the triangle $AIC$. The extension of $BA$ beyond $A$ meets $\\Omega$ at $X$, and the extension of $BC$ beyond $C$ meets $\\Omega$ at $Z$. The extensions of $AD$ and $CD$ beyond $D$ meet $\\Omega$ at $Y$ and $T$, respectively. Prove that \\AD + DT + T X + XA = CD + DY + Y Z + ZC\\", "data_name": "Geometry", "category": "Geometry"}
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{"name": "imo_2022_p4", "split": "test", "informal_prefix": "/--Let $ABCDE$ be a convex pentagon such that $BC = DE$. Assume that there is a point $T$ inside $ABCDE$ with $TB = TD$, $TC = TE$ and $\\angle ABT = \\angle TEA$. Let line $AB$ intersect lines $CD$ and $CT$ at points $P$ and $Q$, respectively. Assume that the points $P, B, A, Q$ occur on their line in that order. Let line $AE$ intersect lines $CD$ and $DT$ at points $R$ and $S$, respectively. Assume that the points $R, E, A, S$ occur on their line in that order. Prove that the points $P, S, Q, R$ lie on a circle.-/\n", "formal_statement": "import SystemE\nimport UniGeo.Relations\nimport UniGeo.Abbre\nimport UniGeo.Theorem\n\n/--Let $ABCDE$ be a convex pentagon such that $BC = DE$. Assume that there is a point $T$ inside $ABCDE$ with $TB = TD$, $TC = TE$ and $\\angle ABT = \\angle TEA$. Let line $AB$ intersect lines $CD$ and $CT$ at points $P$ and $Q$, respectively. Assume that the points $P, B, A, Q$ occur on their line in that order. Let line $AE$ intersect lines $CD$ and $DT$ at points $R$ and $S$, respectively. Assume that the points $R, E, A, S$ occur on their line in that order. Prove that the points $P, S, Q, R$ lie on a circle.-/\ntheorem problem_imo_2022_p4 : ∀ (A B C D T Y T R S: Point) (AB BC CD DE EA EA CT: Line), (formPentagon A B C D E AB BC CD DE EA) ∧\n|(T─B)|=|(T─D)| ∧ |(T─C)|=|(T─E)| ∧ |(T─B)|=|(T─D)| ∧ ∠A:B:T = ∠T:E:A ∧\ntwoLinesIntersectAtPoint AB CD P ∧ distinctPointsOnLine C T CT ∧ twoLinesIntersectAtPoint AB CT Q ∧\nbetween P B A ∧ between B A Q ∧\ndistinctPointsOnLine D T DT ∧\ntwoLinesIntersectAtPoint CD EA R\n∧ twoLinesIntersectAtPoint DT EA S\n∧ between R E A ∧ between E A S\n→ cyclic P S Q R := by\n", "goal": "∀ (A B C D T Y T R S: Point) (AB BC CD DE EA EA CT: Line), (formPentagon A B C D E AB BC CD DE EA) ∧\n|(T─B)|=|(T─D)| ∧ |(T─C)|=|(T─E)| ∧ |(T─B)|=|(T─D)| ∧ ∠A:B:T = ∠T:E:A ∧\ntwoLinesIntersectAtPoint AB CD P ∧ distinctPointsOnLine C T CT ∧ twoLinesIntersectAtPoint AB CT Q ∧\nbetween P B A ∧ between B A Q ∧\ndistinctPointsOnLine D T DT ∧\ntwoLinesIntersectAtPoint CD EA R\n∧ twoLinesIntersectAtPoint DT EA S\n∧ between R E A ∧ between E A S\n→ cyclic P S Q R ", "header": "import SystemE\nimport UniGeo.Relations\nimport UniGeo.Abbre\nimport UniGeo.Theorem\n\n", "formal_proof": "", "uuid": "6ce19205-2d5b-4a41-9071-9ed14e7c80e0", "statementid": "6ce19205-2d5b-4a41-9071-9ed14e7c80e0", "natural_language": "Let $ABCDE$ be a convex pentagon such that $BC = DE$. Assume that there is a point $T$ inside $ABCDE$ with $TB = TD$, $TC = TE$ and $\\angle ABT = \\angle TEA$. Let line $AB$ intersect lines $CD$ and $CT$ at points $P$ and $Q$, respectively. Assume that the points $P, B, A, Q$ occur on their line in that order. Let line $AE$ intersect lines $CD$ and $DT$ at points $R$ and $S$, respectively. Assume that the points $R, E, A, S$ occur on their line in that order. Prove that the points $P, S, Q, R$ lie on a circle.", "data_name": "Geometry", "category": "Geometry"}
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{"name": "imo_2023_p2", "split": "test", "informal_prefix": "/--Let $ABC$ be an acute-angled triangle with $AB < AC$. Let $\\Omega$ be the circumcircle of $ABC$. Let $S$ be the midpoint of the arc $CB$ of $\\Omega$ containing $A$. The perpendicular from $A$ to $BC$ meets $BS$ at $D$ and meets $\\Omega$ again at $E \\neq A$. The line through $D$ parallel to $BC$ meets line $BE$ at $L$. Denote the circumcircle of triangle $BDL$ by $\\omega$. Let $\\omega$ meet $\\Omega$ again at $P \\neq B$. Prove that the line tangent to $\\omega$ at $P$ meets line $BS$ on the internal angle bisector of $\\angle BAC$.-/\n", "formal_statement": "import SystemE\nimport UniGeo.Relations\nimport UniGeo.Abbre\nimport UniGeo.Theorem\n\n/--Let $ABC$ be an acute-angled triangle with $AB < AC$. Let $\\Omega$ be the circumcircle of $ABC$. Let $S$ be the midpoint of the arc $CB$ of $\\Omega$ containing $A$. The perpendicular from $A$ to $BC$ meets $BS$ at $D$ and meets $\\Omega$ again at $E \\neq A$. The line through $D$ parallel to $BC$ meets line $BE$ at $L$. Denote the circumcircle of triangle $BDL$ by $\\omega$. Let $\\omega$ meet $\\Omega$ again at $P \\neq B$. Prove that the line tangent to $\\omega$ at $P$ meets line $BS$ on the internal angle bisector of $\\angle BAC$.-/\ntheorem problem_imo_2023_p2 : ∀ (A B C D E S L P X: Point) (AB BC CA ADE BE DL l BS: Line) (Ω ω: Circle), formAcuteTriangle A B C AB BC CA ∧\n|(A─B)| < |(A─C)| ∧ circumCircle Ω A B C ∧\nS.onCircle Ω ∧ |(S─C)| = |(S─B)| ∧ S.sameSide A BC\n∧ threePointsOnLine A D E ADE ∧ perpLine ADE BC ∧\nD.onLine BS ∧ E.onCircle Ω ∧ threePointsOnLine B E L BE ∧\ndistinctPointsOnLine D L DL ∧ ¬(BE.intersectsLine DL) ∧\ncircumCircle ω B D L ∧\nP.onCircle Ω ∧ P.onCircle ω ∧ P ≠ B ∧\ndistinctPointsOnLine B S BS ∧\ntangentAtPoint l ω P ∧ twoLinesIntersectAtPoint l BS X\n → ∠ X:A:B = ∠X:A:C := by\n", "goal": "∀ (A B C D E S L P X: Point) (AB BC CA ADE BE DL l BS: Line) (Ω ω: Circle), formAcuteTriangle A B C AB BC CA ∧\n|(A─B)| < |(A─C)| ∧ circumCircle Ω A B C ∧\nS.onCircle Ω ∧ |(S─C)| = |(S─B)| ∧ S.sameSide A BC\n∧ threePointsOnLine A D E ADE ∧ perpLine ADE BC ∧\nD.onLine BS ∧ E.onCircle Ω ∧ threePointsOnLine B E L BE ∧\ndistinctPointsOnLine D L DL ∧ ¬(BE.intersectsLine DL) ∧\ncircumCircle ω B D L ∧\nP.onCircle Ω ∧ P.onCircle ω ∧ P ≠ B ∧\ndistinctPointsOnLine B S BS ∧\ntangentAtPoint l ω P ∧ twoLinesIntersectAtPoint l BS X\n → ∠ X:A:B = ∠X:A:C ", "header": "import SystemE\nimport UniGeo.Relations\nimport UniGeo.Abbre\nimport UniGeo.Theorem\n\n", "formal_proof": "", "uuid": "b6b199ab-f482-4313-b2e3-0776ae16c1d1", "statementid": "b6b199ab-f482-4313-b2e3-0776ae16c1d1", "natural_language": "Let $ABC$ be an acute-angled triangle with $AB < AC$. Let $\\Omega$ be the circumcircle of $ABC$. Let $S$ be the midpoint of the arc $CB$ of $\\Omega$ containing $A$. The perpendicular from $A$ to $BC$ meets $BS$ at $D$ and meets $\\Omega$ again at $E \\neq A$. The line through $D$ parallel to $BC$ meets line $BE$ at $L$. Denote the circumcircle of triangle $BDL$ by $\\omega$. Let $\\omega$ meet $\\Omega$ again at $P \\neq B$. Prove that the line tangent to $\\omega$ at $P$ meets line $BS$ on the internal angle bisector of $\\angle BAC$.", "data_name": "Geometry", "category": "Geometry"}
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{"name": "Elements_Prop6", "split": "test", "informal_prefix": "/--Prove that if an triangle ABC such that \\angle ABC = \\angle ACB, then AB=AC.-/\n", "formal_statement": "import SystemE\nimport UniGeo.Relations\nimport UniGeo.Abbre\nimport UniGeo.Theorem\n\n/--Prove that if an triangle ABC such that \\angle ABC = \\angle ACB, then AB=AC.-/\ntheorem problem_Elements_Prop6 : ∀ (a b c : Point) (AB BC AC : Line),\n formTriangle a b c AB BC AC ∧ (∠ a:b:c = ∠ a:c:b) →\n |(a─b)| = |(a─c)| :=\nby\n ", "goal": "∀ (a b c : Point) (AB BC AC : Line),\n formTriangle a b c AB BC AC ∧ (∠ a:b:c = ∠ a:c:b) →\n |(a─b)| = |(a─c)| ", "header": "import SystemE\nimport UniGeo.Relations\nimport UniGeo.Abbre\nimport UniGeo.Theorem\n\n", "formal_proof": "", "uuid": "416711a7-bea8-4baf-847b-a7ca43541f6d", "statementid": "416711a7-bea8-4baf-847b-a7ca43541f6d", "natural_language": "Prove that if an triangle ABC such that \\angle ABC = \\angle ACB, then AB=AC.", "data_name": "Geometry", "category": "Geometry"}
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{"name": "Elements_Prop37", "split": "test", "informal_prefix": "/--ABC, BCD are two triangles and AD is parallel to BC. C,D are on the same side of AB. Prove that the area of triangle ABC is equal to the area of triangle DBC.-/\n", "formal_statement": "import SystemE\nimport UniGeo.Relations\nimport UniGeo.Abbre\nimport UniGeo.Theorem\n\n/--ABC, BCD are two triangles and AD is parallel to BC. C,D are on the same side of AB. Prove that the area of triangle ABC is equal to the area of triangle DBC.-/\ntheorem problem_Elements_Prop37 : ∀ (a b c d : Point) (AB BC AC BD CD AD : Line),\n formTriangle a b c AB BC AC ∧ formTriangle d b c BD BC CD ∧ distinctPointsOnLine a d AD ∧\n ¬(AD.intersectsLine BC) ∧ d.sameSide c AB →\n Triangle.area △ a:b:c = Triangle.area △ d:b:c :=\nby\n ", "goal": "∀ (a b c d : Point) (AB BC AC BD CD AD : Line),\n formTriangle a b c AB BC AC ∧ formTriangle d b c BD BC CD ∧ distinctPointsOnLine a d AD ∧\n ¬(AD.intersectsLine BC) ∧ d.sameSide c AB →\n Triangle.area △ a:b:c = Triangle.area △ d:b:c ", "header": "import SystemE\nimport UniGeo.Relations\nimport UniGeo.Abbre\nimport UniGeo.Theorem\n\n", "formal_proof": "", "uuid": "63a9aa51-1f8e-4fa3-a7fa-fcff7ed1cd44", "statementid": "63a9aa51-1f8e-4fa3-a7fa-fcff7ed1cd44", "natural_language": "ABC, BCD are two triangles and AD is parallel to BC. C,D are on the same side of AB. Prove that the area of triangle ABC is equal to the area of triangle DBC.", "data_name": "Geometry", "category": "Geometry"}
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{"name": "UniGeo_Quadrilateral1", "split": "test", "informal_prefix": "/--In a quadrilateral QRST, QR ⟂ RS , QT ⟂ ST , and ∠ QSR = ∠ SQT. Prove that ST = QR.-/\n", "formal_statement": "import SystemE\nimport UniGeo.Relations\nimport UniGeo.Abbre\nimport UniGeo.Theorem\n\n/--In a quadrilateral QRST, QR ⟂ RS , QT ⟂ ST , and ∠ QSR = ∠ SQT. Prove that ST = QR.-/\ntheorem problem_UniGeo_Quadrilateral1 : ∀ (Q R S T : Point) (QR ST RS QT QS : Line),\n formQuadrilateral Q R T S QR ST QT RS ∧\n distinctPointsOnLine Q S QS ∧\n ∠ Q:R:S = ∟ ∧\n ∠ Q:T:S = ∟ ∧\n ∠ Q:S:R = ∠ S:Q:T →\n |(S─T)| = |(Q─R)| := by\n ", "goal": "∀ (Q R S T : Point) (QR ST RS QT QS : Line),\n formQuadrilateral Q R T S QR ST QT RS ∧\n distinctPointsOnLine Q S QS ∧\n ∠ Q:R:S = ∟ ∧\n ∠ Q:T:S = ∟ ∧\n ∠ Q:S:R = ∠ S:Q:T →\n |(S─T)| = |(Q─R)| ", "header": "import SystemE\nimport UniGeo.Relations\nimport UniGeo.Abbre\nimport UniGeo.Theorem\n\n", "formal_proof": "", "uuid": "c70293a4-1a88-4ae6-bb22-b72118437daa", "statementid": "c70293a4-1a88-4ae6-bb22-b72118437daa", "natural_language": "In a quadrilateral QRST, QR ⟂ RS , QT ⟂ ST , and ∠ QSR = ∠ SQT. Prove that ST = QR.", "data_name": "Geometry", "category": "Geometry"}
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{"name": "UniGeo_Similarity20", "split": "test", "informal_prefix": "/--In triangle HIF, HI ⟂ FI. Given GI ⟂ FH and G is on the line HF, prove that △ F H I ~ △ I H G.-/\n", "formal_statement": "import SystemE\nimport UniGeo.Relations\nimport UniGeo.Abbre\nimport UniGeo.Theorem\n\n/--In triangle HIF, HI ⟂ FI. Given GI ⟂ FH and G is on the line HF, prove that △ F H I ~ △ I H G.-/\ntheorem problem_UniGeo_Similarity20 : ∀ (F G H I : Point) (FH FI HI GI : Line),\n formTriangle F G I FH GI FI ∧\n formTriangle G H I FH HI GI ∧\n between F G H ∧\n ∠ F:G:I = ∟ ∧ ∠ I:G:H = ∟ ∧\n ∠ H:I:F = ∟ →\n (△ F:H:I).similar (△ I:H:G) :=\nby\n ", "goal": "∀ (F G H I : Point) (FH FI HI GI : Line),\n formTriangle F G I FH GI FI ∧\n formTriangle G H I FH HI GI ∧\n between F G H ∧\n ∠ F:G:I = ∟ ∧ ∠ I:G:H = ∟ ∧\n ∠ H:I:F = ∟ →\n (△ F:H:I).similar (△ I:H:G) ", "header": "import SystemE\nimport UniGeo.Relations\nimport UniGeo.Abbre\nimport UniGeo.Theorem\n\n", "formal_proof": "", "uuid": "64c935c5-2501-419b-bf4d-b90b258f01c2", "statementid": "64c935c5-2501-419b-bf4d-b90b258f01c2", "natural_language": "In triangle HIF, HI ⟂ FI. Given GI ⟂ FH and G is on the line HF, prove that △ F H I ~ △ I H G.", "data_name": "Geometry", "category": "Geometry"}
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{"name": "UniGeo_Triangle3", "split": "test", "informal_prefix": "/--In a triangle HIJ, HI = HJ. K is a point on segment IJ such that \\angle IHK = \\angle JHK. Prove that \\angle HKI = \\angle HKJ.-/\n", "formal_statement": "import SystemE\nimport UniGeo.Relations\nimport UniGeo.Abbre\nimport UniGeo.Theorem\n\n/--In a triangle HIJ, HI = HJ. K is a point on segment IJ such that \\angle IHK = \\angle JHK. Prove that \\angle HKI = \\angle HKJ.-/\ntheorem problem_UniGeo_Triangle3 : ∀ (H I J K : Point) (HI IJ JH HK : Line),\n formTriangle H I J HI IJ JH ∧\n distinctPointsOnLine H K HK ∧\n IJ.intersectsLine HK ∧ K.onLine IJ ∧ between I K J ∧\n ∠ I:H:K = ∠ J:H:K ∧\n |(H─I)| = |(H─J)| →\n ∠ H:K:I = ∠ H:K:J :=\nby\n ", "goal": "∀ (H I J K : Point) (HI IJ JH HK : Line),\n formTriangle H I J HI IJ JH ∧\n distinctPointsOnLine H K HK ∧\n IJ.intersectsLine HK ∧ K.onLine IJ ∧ between I K J ∧\n ∠ I:H:K = ∠ J:H:K ∧\n |(H─I)| = |(H─J)| →\n ∠ H:K:I = ∠ H:K:J ", "header": "import SystemE\nimport UniGeo.Relations\nimport UniGeo.Abbre\nimport UniGeo.Theorem\n\n", "formal_proof": "", "uuid": "a162d2ca-24ce-4476-99a9-f51955b75608", "statementid": "a162d2ca-24ce-4476-99a9-f51955b75608", "natural_language": "In a triangle HIJ, HI = HJ. K is a point on segment IJ such that \\angle IHK = \\angle JHK. Prove that \\angle HKI = \\angle HKJ.", "data_name": "Geometry", "category": "Geometry"}
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{"name": "HP97", "split": "test", "informal_prefix": "/--Given that in $\\triangle ABC$, $O$ is the circumcenter, and $I$ is the incenter. Line $OI$ is perpendicular to line $AI$. Prove that $AB + AC = 2BC$.-/\n", "formal_statement": "import SystemE\nimport UniGeo.Relations\nimport UniGeo.Abbre\nimport UniGeo.Theorem\n\n/--Given that in $\\triangle ABC$, $O$ is the circumcenter, and $I$ is the incenter. Line $OI$ is perpendicular to line $AI$. Prove that $AB + AC = 2BC$.-/\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", "goal": " ∀ (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)|) ", "header": "import SystemE\nimport UniGeo.Relations\nimport UniGeo.Abbre\nimport UniGeo.Theorem\n\n", "formal_proof": "", "uuid": "bee0a966-2427-45ef-ac8b-36cf4535ae05", "statementid": "bee0a966-2427-45ef-ac8b-36cf4535ae05", "natural_language": "Given that in $\\triangle ABC$, $O$ is the circumcenter, and $I$ is the incenter. Line $OI$ is perpendicular to line $AI$. Prove that $AB + AC = 2BC$.", "data_name": "Geometry", "category": "Geometry"}
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| 12 |
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{"name": "HP71", "split": "test", "informal_prefix": "/--Given that in $\\triangle ABC$, lines $AD$, $BE$, and $CF$ are the altitudes of $\\triangle ABC$. Point $H$ is the orthocenter of $\\triangle ABC$, and point $O$ is the circumcenter of $\\triangle ABC$. Line $ED$ intersects line $AB$ at point $M$, and line $FD$ intersects line $AC$ at point $N$. Prove that $OH \\perp MN$.-/\n", "formal_statement": "import SystemE\nimport UniGeo.Relations\nimport UniGeo.Abbre\nimport UniGeo.Theorem\n\n/--Given that in $\\triangle ABC$, lines $AD$, $BE$, and $CF$ are the altitudes of $\\triangle ABC$. Point $H$ is the orthocenter of $\\triangle ABC$, and point $O$ is the circumcenter of $\\triangle ABC$. Line $ED$ intersects line $AB$ at point $M$, and line $FD$ intersects line $AC$ at point $N$. Prove that $OH \\perp MN$.-/\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 ", "goal": " ∀ (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", "header": "import SystemE\nimport UniGeo.Relations\nimport UniGeo.Abbre\nimport UniGeo.Theorem\n\n", "formal_proof": "", "uuid": "7419cc04-18ed-448e-b7a9-5dc81706f352", "statementid": "7419cc04-18ed-448e-b7a9-5dc81706f352", "natural_language": "Given that in $\\triangle ABC$, lines $AD$, $BE$, and $CF$ are the altitudes of $\\triangle ABC$. Point $H$ is the orthocenter of $\\triangle ABC$, and point $O$ is the circumcenter of $\\triangle ABC$. Line $ED$ intersects line $AB$ at point $M$, and line $FD$ intersects line $AC$ at point $N$. Prove that $OH \\perp MN$.", "data_name": "Geometry", "category": "Geometry"}
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{"name": "MP97", "split": "test", "informal_prefix": "/--In $\\triangle ABC$, points $D$ and $E$ are on sides $AB$ and $AC$, respectively, such that $BD = CE$. Lines $BE$ and $CD$ intersect at point $O$, and $OD = OE$. Prove that $AB = AC$.-/\n", "formal_statement": "import SystemE\nimport UniGeo.Relations\nimport UniGeo.Abbre\nimport UniGeo.Theorem\n\n/--In $\\triangle ABC$, points $D$ and $E$ are on sides $AB$ and $AC$, respectively, such that $BD = CE$. Lines $BE$ and $CD$ intersect at point $O$, and $OD = OE$. Prove that $AB = AC$.-/\ntheorem problem_MP97 :\n∀ (A B C D E F G : Point) (AB BC CD DA AC AD EF : Line),\n (formQuadrilateral A B C D AB BC CD DA)\n ∧ (∠B:A:D = ∠A:D:C)\n ∧ (E.onLine BC)\n ∧ (between B E C)\n ∧ (|(B─E)| = |(A─B)|)\n ∧ (|(C─E)| = |(C─D)|)\n ∧ (¬(EF.intersectsLine AB))\n ∧ distinctPointsOnLine E F EF\n ∧ distinctPointsOnLine A C AC\n ∧ (twoLinesIntersectAtPoint EF AC G)\n ∧ (twoLinesIntersectAtPoint EF AD F)\n → (|(E─G)| = |(G─F)|)\n:= by\n ", "goal": "∀ (A B C D E F G : Point) (AB BC CD DA AC AD EF : Line),\n (formQuadrilateral A B C D AB BC CD DA)\n ∧ (∠B:A:D = ∠A:D:C)\n ∧ (E.onLine BC)\n ∧ (between B E C)\n ∧ (|(B─E)| = |(A─B)|)\n ∧ (|(C─E)| = |(C─D)|)\n ∧ (¬(EF.intersectsLine AB))\n ∧ distinctPointsOnLine E F EF\n ∧ distinctPointsOnLine A C AC\n ∧ (twoLinesIntersectAtPoint EF AC G)\n ∧ (twoLinesIntersectAtPoint EF AD F)\n → (|(E─G)| = |(G─F)|)\n", "header": "import SystemE\nimport UniGeo.Relations\nimport UniGeo.Abbre\nimport UniGeo.Theorem\n\n", "formal_proof": "", "uuid": "befcd99f-0f90-41bb-997a-596035955d64", "statementid": "befcd99f-0f90-41bb-997a-596035955d64", "natural_language": "In $\\triangle ABC$, points $D$ and $E$ are on sides $AB$ and $AC$, respectively, such that $BD = CE$. Lines $BE$ and $CD$ intersect at point $O$, and $OD = OE$. Prove that $AB = AC$.", "data_name": "Geometry", "category": "Geometry"}
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{"name": "MP19", "split": "test", "informal_prefix": "/--In quadrilateral $ABCD$, $AE \\perp BD$ at point $E$, $CF \\perp BD$ at point $F$, and point $P$ is the midpoint of $AC$. Prove that $PE = PF$.-/\n", "formal_statement": "import SystemE\nimport UniGeo.Relations\nimport UniGeo.Abbre\nimport UniGeo.Theorem\n\n/--In quadrilateral $ABCD$, $AE \\perp BD$ at point $E$, $CF \\perp BD$ at point $F$, and point $P$ is the midpoint of $AC$. Prove that $PE = PF$.-/\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 ", "goal": "∀ (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)| ", "header": "import SystemE\nimport UniGeo.Relations\nimport UniGeo.Abbre\nimport UniGeo.Theorem\n\n", "formal_proof": "", "uuid": "6ef3fbb0-6b67-44d4-82a0-63c6fd65b96b", "statementid": "6ef3fbb0-6b67-44d4-82a0-63c6fd65b96b", "natural_language": "In quadrilateral $ABCD$, $AE \\perp BD$ at point $E$, $CF \\perp BD$ at point $F$, and point $P$ is the midpoint of $AC$. Prove that $PE = PF$.", "data_name": "Geometry", "category": "Geometry"}
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{"name": "CO20", "split": "test", "informal_prefix": "/--As shown in the figure, in an isosceles triangle $ABC$ with $AB = BC$, $I$ is the incenter, $M$ is the midpoint of $BI$, $P$ is a point on side $AC$ such that $AP = 3PC$, and $PI$ is extended to a point $H$ such that $MH \\perp PH$. $Q$ is the midpoint of $AB$ on the circumcircle of $\\triangle ABC$. Prove: $BH \\perp QH$.-/\n", "formal_statement": "import SystemE\nimport UniGeo.Relations\nimport UniGeo.Abbre\nimport UniGeo.Theorem\n\n/--As shown in the figure, in an isosceles triangle $ABC$ with $AB = BC$, $I$ is the incenter, $M$ is the midpoint of $BI$, $P$ is a point on side $AC$ such that $AP = 3PC$, and $PI$ is extended to a point $H$ such that $MH \\perp PH$. $Q$ is the midpoint of $AB$ on the circumcircle of $\\triangle ABC$. Prove: $BH \\perp QH$.-/\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 ", "goal": "∀ (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 ", "header": "import SystemE\nimport UniGeo.Relations\nimport UniGeo.Abbre\nimport UniGeo.Theorem\n\n", "formal_proof": "", "uuid": "16085ff7-9750-43b2-85fc-1e953dfd7659", "statementid": "16085ff7-9750-43b2-85fc-1e953dfd7659", "natural_language": "As shown in the figure, in an isosceles triangle $ABC$ with $AB = BC$, $I$ is the incenter, $M$ is the midpoint of $BI$, $P$ is a point on side $AC$ such that $AP = 3PC$, and $PI$ is extended to a point $H$ such that $MH \\perp PH$. $Q$ is the midpoint of $AB$ on the circumcircle of $\\triangle ABC$. Prove: $BH \\perp QH$.", "data_name": "Geometry", "category": "Geometry"}
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{"name": "CO19", "split": "test", "informal_prefix": "/--As shown in the figure, in an acute-angled triangle $ABC$, $M$ is the midpoint of side $BC$, and point $P$ lies on side $AB$ such that $AP$ bisects $\\angle BAC$. The line $MP$ intersects the excircles of $\\triangle ABP$ and $\\triangle ACP$ at points $D$ and $E$, respectively. Prove: If $DE = MP$, then $BC = 2BP$.-/\n", "formal_statement": "import SystemE\nimport UniGeo.Relations\nimport UniGeo.Abbre\nimport UniGeo.Theorem\n\n/--As shown in the figure, in an acute-angled triangle $ABC$, $M$ is the midpoint of side $BC$, and point $P$ lies on side $AB$ such that $AP$ bisects $\\angle BAC$. The line $MP$ intersects the excircles of $\\triangle ABP$ and $\\triangle ACP$ at points $D$ and $E$, respectively. Prove: If $DE = MP$, then $BC = 2BP$.-/\ntheorem problem_CO19 :\n ∀ (A B C M P D E : Point) (AB BC CA AP MP : Line) (ω₁ ω₂ : Circle),\n formAcuteTriangle A B C AB BC CA ∧\n midpoint M B C ∧\n insideTriangle P A B C AB BC CA ∧\n distinctPointsOnLine A P AP ∧\n ∠B:A:P=∠C:A:P ∧\n distinctPointsOnLine M P MP ∧\n circumCircle ω₁ A B P ∧\n circumCircle ω₂ A C P ∧\n D.onLine MP ∧ D.onCircle ω₁ ∧\n E.onLine MP ∧ E.onCircle ω₂ ∧\n |(D─E)| = |(M─P)|\n → (|(B─C)| = 2 * |(B─P)|) := by\n ", "goal": " ∀ (A B C M P D E : Point) (AB BC CA AP MP : Line) (ω₁ ω₂ : Circle),\n formAcuteTriangle A B C AB BC CA ∧\n midpoint M B C ∧\n insideTriangle P A B C AB BC CA ∧\n distinctPointsOnLine A P AP ∧\n ∠B:A:P=∠C:A:P ∧\n distinctPointsOnLine M P MP ∧\n circumCircle ω₁ A B P ∧\n circumCircle ω₂ A C P ∧\n D.onLine MP ∧ D.onCircle ω₁ ∧\n E.onLine MP ∧ E.onCircle ω₂ ∧\n |(D─E)| = |(M─P)|\n → (|(B─C)| = 2 * |(B─P)|) ", "header": "import SystemE\nimport UniGeo.Relations\nimport UniGeo.Abbre\nimport UniGeo.Theorem\n\n", "formal_proof": "", "uuid": "617a2c10-e1a0-4c52-82bf-e2895e0f9e8d", "statementid": "617a2c10-e1a0-4c52-82bf-e2895e0f9e8d", "natural_language": "As shown in the figure, in an acute-angled triangle $ABC$, $M$ is the midpoint of side $BC$, and point $P$ lies on side $AB$ such that $AP$ bisects $\\angle BAC$. The line $MP$ intersects the excircles of $\\triangle ABP$ and $\\triangle ACP$ at points $D$ and $E$, respectively. Prove: If $DE = MP$, then $BC = 2BP$.", "data_name": "Geometry", "category": "Geometry"}
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{"name": "MP9", "split": "test", "informal_prefix": "/--In $\\triangle ABC$, $\\angle A = 3\\angle C$ and $BC = 2AB$. Prove that $\\angle BAC = 90^\\circ$.-/\n", "formal_statement": "import SystemE\nimport UniGeo.Relations\nimport UniGeo.Abbre\nimport UniGeo.Theorem\n\n/--In $\\triangle ABC$, $\\angle A = 3\\angle C$ and $BC = 2AB$. Prove that $\\angle BAC = 90^\\circ$.-/\ntheorem problem_MP97 :\n∀ (A B C D E F G : Point) (AB BC CD DA AC AD EF : Line),\n (formQuadrilateral A B C D AB BC CD DA)\n ∧ (∠B:A:D = ∠A:D:C)\n ∧ (E.onLine BC)\n ∧ (between B E C)\n ∧ (|(B─E)| = |(A─B)|)\n ∧ (|(C─E)| = |(C─D)|)\n ∧ (¬(EF.intersectsLine AB))\n ∧ distinctPointsOnLine E F EF\n ∧ distinctPointsOnLine A C AC\n ∧ (twoLinesIntersectAtPoint EF AC G)\n ∧ (twoLinesIntersectAtPoint EF AD F)\n → (|(E─G)| = |(G─F)|)\n:= by\n ", "goal": "\n∀ (A B C D E F G : Point) (AB BC CD DA AC AD EF : Line),\n (formQuadrilateral A B C D AB BC CD DA)\n ∧ (∠B:A:D = ∠A:D:C)\n ∧ (E.onLine BC)\n ∧ (between B E C)\n ∧ (|(B─E)| = |(A─B)|)\n ∧ (|(C─E)| = |(C─D)|)\n ∧ (¬(EF.intersectsLine AB))\n ∧ distinctPointsOnLine E F EF\n ∧ distinctPointsOnLine A C AC\n ∧ (twoLinesIntersectAtPoint EF AC G)\n ∧ (twoLinesIntersectAtPoint EF AD F)\n → (|(E─G)| = |(G─F)|)\n", "header": "import SystemE\nimport UniGeo.Relations\nimport UniGeo.Abbre\nimport UniGeo.Theorem\n\n", "formal_proof": "", "uuid": "2d22756f-b7f4-4169-a8c2-ce203e122d3e", "statementid": "2d22756f-b7f4-4169-a8c2-ce203e122d3e", "natural_language": "In $\\triangle ABC$, $\\angle A = 3\\angle C$ and $BC = 2AB$. Prove that $\\angle BAC = 90^\\circ$.", "data_name": "Geometry", "category": "Geometry"}
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{"name": "MP75", "split": "test", "informal_prefix": "/--In convex quadrilateral $ABCD$, diagonals $AC$ and $BD$ intersect at point $O$, $AB + AC = DB + DC$, and $OB > OC$. Prove that $OA > OD$.-/\n", "formal_statement": "import SystemE\nimport UniGeo.Relations\nimport UniGeo.Abbre\nimport UniGeo.Theorem\n\n/--In convex quadrilateral $ABCD$, diagonals $AC$ and $BD$ intersect at point $O$, $AB + AC = DB + DC$, and $OB > OC$. Prove that $OA > OD$.-/\ntheorem problem_MP75 :\n ∀ (A B C D O : Point) (AB BC CD DA AC BD : Line),\n formQuadrilateral A B C D AB BC CD DA ∧\n distinctPointsOnLine A C AC ∧\n distinctPointsOnLine B D BD ∧\n twoLinesIntersectAtPoint AC BD O ∧\n (|(A─B)| + |(A─C)| = |(D─B)| + |(D─C)|) ∧\n (|(O─B)| > |(O─C)|)\n → (|(O─A)| > |(O─D)|) := by\n ", "goal": " ∀ (A B C D O : Point) (AB BC CD DA AC BD : Line),\n formQuadrilateral A B C D AB BC CD DA ∧\n distinctPointsOnLine A C AC ∧\n distinctPointsOnLine B D BD ∧\n twoLinesIntersectAtPoint AC BD O ∧\n (|(A─B)| + |(A─C)| = |(D─B)| + |(D─C)|) ∧\n (|(O─B)| > |(O─C)|)\n → (|(O─A)| > |(O─D)|) ", "header": "import SystemE\nimport UniGeo.Relations\nimport UniGeo.Abbre\nimport UniGeo.Theorem\n\n", "formal_proof": "", "uuid": "82b962af-2493-42cb-8a8d-3afb9c57a6cb", "statementid": "82b962af-2493-42cb-8a8d-3afb9c57a6cb", "natural_language": "In convex quadrilateral $ABCD$, diagonals $AC$ and $BD$ intersect at point $O$, $AB + AC = DB + DC$, and $OB > OC$. Prove that $OA > OD$.", "data_name": "Geometry", "category": "Geometry"}
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{"name": "MP12", "split": "test", "informal_prefix": "/--In $\\triangle ABC$, $AB = AC$, $\\angle A = 20^\\circ$, and the angle bisector of $\\angle ABC$ intersects $AC$ at point $D$. Prove that $AD = BD + BC$.-/\n", "formal_statement": "import SystemE\nimport UniGeo.Relations\nimport UniGeo.Abbre\nimport UniGeo.Theorem\n\n/--In $\\triangle ABC$, $AB = AC$, $\\angle A = 20^\\circ$, and the angle bisector of $\\angle ABC$ intersects $AC$ at point $D$. Prove that $AD = BD + BC$.-/\ntheorem problem_MP12 :\n∀ (A B C D : Point) (AB BC CA : Line),\n (formTriangle A B C AB BC CA)\n ∧ (|(A─B)| = |(A─C)|)\n ∧ (∠B:A:C = 2/9 * ∟)\n ∧ D.onLine CA\n ∧ between A D C\n ∧ (∠A:B:D = ∠D:B:C)\n → |(A─D)| = (|(B─D)| + |(B─C)|)\n:= by\n ", "goal": "∀ (A B C D : Point) (AB BC CA : Line),\n (formTriangle A B C AB BC CA)\n ∧ (|(A─B)| = |(A─C)|)\n ∧ (∠B:A:C = 2/9 * ∟)\n ∧ D.onLine CA\n ∧ between A D C\n ∧ (∠A:B:D = ∠D:B:C)\n → |(A─D)| = (|(B─D)| + |(B─C)|)\n", "header": "import SystemE\nimport UniGeo.Relations\nimport UniGeo.Abbre\nimport UniGeo.Theorem\n\n", "formal_proof": "", "uuid": "7277f964-564f-4251-a62d-278e7b8cba0d", "statementid": "7277f964-564f-4251-a62d-278e7b8cba0d", "natural_language": "In $\\triangle ABC$, $AB = AC$, $\\angle A = 20^\\circ$, and the angle bisector of $\\angle ABC$ intersects $AC$ at point $D$. Prove that $AD = BD + BC$.", "data_name": "Geometry", "category": "Geometry"}
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| 20 |
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{"name": "MP10", "split": "test", "informal_prefix": "/--In $\\triangle ABC$, $AB = AC$, and $\\angle A = 30^\\circ$. Prove that $\\triangle ABC$ is an equilateral triangle.-/\n", "formal_statement": "import SystemE\nimport UniGeo.Relations\nimport UniGeo.Abbre\nimport UniGeo.Theorem\n\n/--In $\\triangle ABC$, $AB = AC$, and $\\angle A = 30^\\circ$. Prove that $\\triangle ABC$ is an equilateral triangle.-/\ntheorem problem_MP10 : ∀\n(A B C D : Point)\n(AB BC CA : Line),\nformTriangle A B C AB BC CA ∧\n|(A─B)| = |(A─C)| ∧\nmidpoint A D C ∧\n(∠A:B:D * 3= ∟ )\n→ (|(A─B)|=|(B─C)| ∧ |(B─C)|=|(C─A)|) := by\n", "goal": "∀\n(A B C D : Point)\n(AB BC CA : Line),\nformTriangle A B C AB BC CA ∧\n|(A─B)| = |(A─C)| ∧\nmidpoint A D C ∧\n(∠A:B:D * 3= ∟ )\n→ (|(A─B)|=|(B─C)| ∧ |(B─C)|=|(C─A)|) ", "header": "import SystemE\nimport UniGeo.Relations\nimport UniGeo.Abbre\nimport UniGeo.Theorem\n\n", "formal_proof": "", "uuid": "e5a34d26-595e-4b73-b9bb-198ff2243d6e", "statementid": "e5a34d26-595e-4b73-b9bb-198ff2243d6e", "natural_language": "In $\\triangle ABC$, $AB = AC$, and $\\angle A = 30^\\circ$. Prove that $\\triangle ABC$ is an equilateral triangle.", "data_name": "Geometry", "category": "Geometry"}
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{"name": "CMO24", "split": "test", "informal_prefix": "/--In triangle $ ABC $, $ I $ is the incenter, and points $ L, M, N $ are the midpoints of segments $ AI, AC, CI $, respectively. Point $ D $ lies on line segment $ AM $, such that $ BC = BD $. In triangle $ ABD $, the incircle touches $ AD, BD $ at points $ E, F $, respectively. Point $ J $ is the circumcenter of triangle $ AIC $, and $ \\omega $ is the circumcircle of triangle $ JMD $. Line $ MN $ intersects $ \\omega $ again at point $ Q $. Prove: The lines $ PQ, LN, EF $ are concurrent-/\n", "formal_statement": "import SystemE\nimport UniGeo.Relations\nimport UniGeo.Abbre\nimport UniGeo.Theorem\n\n/--In triangle $ ABC $, $ I $ is the incenter, and points $ L, M, N $ are the midpoints of segments $ AI, AC, CI $, respectively. Point $ D $ lies on line segment $ AM $, such that $ BC = BD $. In triangle $ ABD $, the incircle touches $ AD, BD $ at points $ E, F $, respectively. Point $ J $ is the circumcenter of triangle $ AIC $, and $ \\omega $ is the circumcircle of triangle $ JMD $. Line $ MN $ intersects $ \\omega $ again at point $ Q $. Prove: The lines $ PQ, LN, EF $ are concurrent-/\ntheorem problem_CMO24 :\n ∀ (A B C I L M N D P Q: Point) (AB BC CA AI AC CI AM BD MN JL EF PQ LN: Line),\n formTriangle A B C AB BC CA ∧\n inCentre I A B C ∧\n midpoint A L I ∧\n midpoint C M A ∧\n midpoint I N C ∧\n between A D M ∧\n distinctPointsOnLine B D BD ∧\n |(B─C)| = |(B─D)| ∧\n inCircle ω AB CA BD ∧\n tangentAtPoint AC ω E ∧\n tangentAtPoint BD ω F ∧\n circumCentre J A I C ∧\n circumCircle ω J M D ∧\n distinctPointsOnLine M N MN ∧\n P.onLine MN ∧\n P.onCircle ω ∧\n distinctPointsOnLine J L JL ∧\n Q.onLine JL ∧\n Q.onCircle ω ∧\n distinctPointsOnLine E F EF ∧\n distinctPointsOnLine L N LN ∧\n distinctPointsOnLine P Q PQ\n → concurrent PQ LN EF := by\n ", "goal": " ∀ (A B C I L M N D P Q: Point) (AB BC CA AI AC CI AM BD MN JL EF PQ LN: Line),\n formTriangle A B C AB BC CA ∧\n inCentre I A B C ∧\n midpoint A L I ∧\n midpoint C M A ∧\n midpoint I N C ∧\n between A D M ∧\n distinctPointsOnLine B D BD ∧\n |(B─C)| = |(B─D)| ∧\n inCircle ω AB CA BD ∧\n tangentAtPoint AC ω E ∧\n tangentAtPoint BD ω F ∧\n circumCentre J A I C ∧\n circumCircle ω J M D ∧\n distinctPointsOnLine M N MN ∧\n P.onLine MN ∧\n P.onCircle ω ∧\n distinctPointsOnLine J L JL ∧\n Q.onLine JL ∧\n Q.onCircle ω ∧\n distinctPointsOnLine E F EF ∧\n distinctPointsOnLine L N LN ∧\n distinctPointsOnLine P Q PQ\n → concurrent PQ LN EF ", "header": "import SystemE\nimport UniGeo.Relations\nimport UniGeo.Abbre\nimport UniGeo.Theorem\n\n", "formal_proof": "", "uuid": "608766e6-8fb4-492d-bb3d-23bcafeed6b7", "statementid": "608766e6-8fb4-492d-bb3d-23bcafeed6b7", "natural_language": "In triangle $ ABC $, $ I $ is the incenter, and points $ L, M, N $ are the midpoints of segments $ AI, AC, CI $, respectively. Point $ D $ lies on line segment $ AM $, such that $ BC = BD $. In triangle $ ABD $, the incircle touches $ AD, BD $ at points $ E, F $, respectively. Point $ J $ is the circumcenter of triangle $ AIC $, and $ \\omega $ is the circumcircle of triangle $ JMD $. Line $ MN $ intersects $ \\omega $ again at point $ Q $. Prove: The lines $ PQ, LN, EF $ are concurrent", "data_name": "Geometry", "category": "Geometry"}
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{"name": "HP13", "split": "test", "informal_prefix": "/--Given that in $\\triangle ABC$, points $D$ and $E$ lie on sides $AB$ and $AC$, respectively, such that $DE \\parallel BC$. Lines $BE$ and $CD$ intersect at point $F$. The circumcircle of $\\triangle BDF$ is $\\odot O$, and the circumcircle of $\\triangle CEF$ is $\\odot P$. These two circles intersect at point $G$. Prove that $\\angle BAF = \\angle CAG$.-/\n", "formal_statement": "import SystemE\nimport UniGeo.Relations\nimport UniGeo.Abbre\nimport UniGeo.Theorem\n\n/--Given that in $\\triangle ABC$, points $D$ and $E$ lie on sides $AB$ and $AC$, respectively, such that $DE \\parallel BC$. Lines $BE$ and $CD$ intersect at point $F$. The circumcircle of $\\triangle BDF$ is $\\odot O$, and the circumcircle of $\\triangle CEF$ is $\\odot P$. These two circles intersect at point $G$. Prove that $\\angle BAF = \\angle CAG$.-/\ntheorem problem_HP13 :\n ∀ (A B C D E F G : Point) (AB BC CA DE BE CD : Line) (O P : Circle),\n formTriangle A B C AB BC CA ∧\n D.onLine AB ∧ between A D B ∧\n E.onLine AC ∧ between A E C ∧\n distinctPointsOnLine D E DE ∧\n ¬(DE.intersectsLine BC) ∧\n distinctPointsOnLine B E BE ∧\n distinctPointsOnLine C D CD ∧\n twoLinesIntersectAtPoint BE CD F ∧\n circumCircle O B D F ∧\n circumCircle P C E F ∧\n G.onCircle O ∧\n G.onCircle P ∧ G ≠ F\n → ∠B:A:F = ∠C:A:G\n:= by\n ", "goal": " ∀ (A B C D E F G : Point) (AB BC CA DE BE CD : Line) (O P : Circle),\n formTriangle A B C AB BC CA ∧\n D.onLine AB ∧ between A D B ∧\n E.onLine AC ∧ between A E C ∧\n distinctPointsOnLine D E DE ∧\n ¬(DE.intersectsLine BC) ∧\n distinctPointsOnLine B E BE ∧\n distinctPointsOnLine C D CD ∧\n twoLinesIntersectAtPoint BE CD F ∧\n circumCircle O B D F ∧\n circumCircle P C E F ∧\n G.onCircle O ∧\n G.onCircle P ∧ G ≠ F\n → ∠B:A:F = ∠C:A:G\n", "header": "import SystemE\nimport UniGeo.Relations\nimport UniGeo.Abbre\nimport UniGeo.Theorem\n\n", "formal_proof": "", "uuid": "cbbdd601-1f51-4275-b786-f2ee413f8b18", "statementid": "cbbdd601-1f51-4275-b786-f2ee413f8b18", "natural_language": "Given that in $\\triangle ABC$, points $D$ and $E$ lie on sides $AB$ and $AC$, respectively, such that $DE \\parallel BC$. Lines $BE$ and $CD$ intersect at point $F$. The circumcircle of $\\triangle BDF$ is $\\odot O$, and the circumcircle of $\\triangle CEF$ is $\\odot P$. These two circles intersect at point $G$. Prove that $\\angle BAF = \\angle CAG$.", "data_name": "Geometry", "category": "Geometry"}
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{"name": "MP35", "split": "test", "informal_prefix": "/--In convex quadrilateral $ABCD$, $BC = CD$, $\\angle ADB = 30^\\circ$, and $\\angle ABD - \\angle DBC = 60^\\circ$. Find $\\angle CAD$.-/\n", "formal_statement": "import SystemE\nimport UniGeo.Relations\nimport UniGeo.Abbre\nimport UniGeo.Theorem\n\n/--In convex quadrilateral $ABCD$, $BC = CD$, $\\angle ADB = 30^\\circ$, and $\\angle ABD - \\angle DBC = 60^\\circ$. Find $\\angle CAD$.-/\ntheorem problem_MP35 : ∀\n (A B C D : Point)\n (AB BC CD DA : Line),\n formQuadrilateral A B C D AB BC CD DA ∧\n |(B─C)| = |(C─D)| ∧\n ∠A:D:B = 1/3 * ∟ ∧\n ∠A:B:D = ∠D:B:C + 2/3 * ∟\n → ∠C:A:D = 1/3 * ∟ := by\n", "goal": "∀\n (A B C D : Point)\n (AB BC CD DA : Line),\n formQuadrilateral A B C D AB BC CD DA ∧\n |(B─C)| = |(C─D)| ∧\n ∠A:D:B = 1/3 * ∟ ∧\n ∠A:B:D = ∠D:B:C + 2/3 * ∟\n → ∠C:A:D = 1/3 * ∟ ", "header": "import SystemE\nimport UniGeo.Relations\nimport UniGeo.Abbre\nimport UniGeo.Theorem\n\n", "formal_proof": "", "uuid": "fb559604-f813-4a93-a007-d112a185afe1", "statementid": "fb559604-f813-4a93-a007-d112a185afe1", "natural_language": "In convex quadrilateral $ABCD$, $BC = CD$, $\\angle ADB = 30^\\circ$, and $\\angle ABD - \\angle DBC = 60^\\circ$. Find $\\angle CAD$.", "data_name": "Geometry", "category": "Geometry"}
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{"name": "MP22", "split": "test", "informal_prefix": "/--In equilateral triangle $\\triangle ABC$, point $D$ is on side $AB$, and point $E$ is on the extension of $CB$ such that $AD = BE$. Prove that $CD = ED$.-/\n", "formal_statement": "import SystemE\nimport UniGeo.Relations\nimport UniGeo.Abbre\nimport UniGeo.Theorem\n\n/--In equilateral triangle $\\triangle ABC$, point $D$ is on side $AB$, and point $E$ is on the extension of $CB$ such that $AD = BE$. Prove that $CD = ED$.-/\ntheorem problem_MP22 :\n∀ (A B C D E : Point) (AB BC CA : Line),\n (formTriangle A B C AB BC CA)\n ∧ (|(A─B)| = |(B─C)|)\n ∧ (|(B─C)| = |(C─A)|)\n ∧ between A D B\n ∧ between C B E\n ∧ (|(A─D)| = |(B─E)|)\n → (|(C─D)| = |(E─D)|) := by\n ", "goal": "∀ (A B C D E : Point) (AB BC CA : Line),\n (formTriangle A B C AB BC CA)\n ∧ (|(A─B)| = |(B─C)|)\n ∧ (|(B─C)| = |(C─A)|)\n ∧ between A D B\n ∧ between C B E\n ∧ (|(A─D)| = |(B─E)|)\n → (|(C─D)| = |(E─D)|) ", "header": "import SystemE\nimport UniGeo.Relations\nimport UniGeo.Abbre\nimport UniGeo.Theorem\n\n", "formal_proof": "", "uuid": "29428ba1-e23a-419f-adcc-5e1d51f1c985", "statementid": "29428ba1-e23a-419f-adcc-5e1d51f1c985", "natural_language": "In equilateral triangle $\\triangle ABC$, point $D$ is on side $AB$, and point $E$ is on the extension of $CB$ such that $AD = BE$. Prove that $CD = ED$.", "data_name": "Geometry", "category": "Geometry"}
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{"name": "HP23", "split": "test", "informal_prefix": "/--Given that $O$ is the circumcenter of $\\triangle ABC$, and points $D$ and $E$ lie on sides $AB$ and $AC$, respectively. Line $OF$ is perpendicular to line $DE$ at point $F$. Points $L$, $M$, and $N$ are the midpoints of line segments $DE$, $BE$, and $CD$, respectively. Prove that points $F$, $L$, $M$, and $N$ are concyclic.-/\n", "formal_statement": "import SystemE\nimport UniGeo.Relations\nimport UniGeo.Abbre\nimport UniGeo.Theorem\n\n/--Given that $O$ is the circumcenter of $\\triangle ABC$, and points $D$ and $E$ lie on sides $AB$ and $AC$, respectively. Line $OF$ is perpendicular to line $DE$ at point $F$. Points $L$, $M$, and $N$ are the midpoints of line segments $DE$, $BE$, and $CD$, respectively. Prove that points $F$, $L$, $M$, and $N$ are concyclic.-/\ntheorem problem_HP23 :\n∀ (A B C O D E F L M N : Point) (AB BC CA DE OF : Line),\n formTriangle A B C AB BC CA ∧\n circumCentre O A B C ∧\n between A D B ∧\n between A E C ∧\n distinctPointsOnLine D E DE ∧\n distinctPointsOnLine O F OF ∧\n twoLinesIntersectAtPoint OF DE F ∧\n perpLine OF DE ∧\n midpoint D L E ∧\n midpoint B M E ∧\n midpoint C N D\n → cyclic F L M N\n:= by\n ", "goal": "∀ (A B C O D E F L M N : Point) (AB BC CA DE OF : Line),\n formTriangle A B C AB BC CA ∧\n circumCentre O A B C ∧\n between A D B ∧\n between A E C ∧\n distinctPointsOnLine D E DE ∧\n distinctPointsOnLine O F OF ∧\n twoLinesIntersectAtPoint OF DE F ∧\n perpLine OF DE ∧\n midpoint D L E ∧\n midpoint B M E ∧\n midpoint C N D\n → cyclic F L M N\n", "header": "import SystemE\nimport UniGeo.Relations\nimport UniGeo.Abbre\nimport UniGeo.Theorem\n\n", "formal_proof": "", "uuid": "f9bf5894-4de0-48f2-ae8b-d4b1e808e12a", "statementid": "f9bf5894-4de0-48f2-ae8b-d4b1e808e12a", "natural_language": "Given that $O$ is the circumcenter of $\\triangle ABC$, and points $D$ and $E$ lie on sides $AB$ and $AC$, respectively. Line $OF$ is perpendicular to line $DE$ at point $F$. Points $L$, $M$, and $N$ are the midpoints of line segments $DE$, $BE$, and $CD$, respectively. Prove that points $F$, $L$, $M$, and $N$ are concyclic.", "data_name": "Geometry", "category": "Geometry"}
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