File size: 2,446 Bytes
d5b7741 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 | import OAI.Combinatorics.Crossing.CircleChords
import OAI.Combinatorics.Crossing.SegmentContinuity
namespace OAI
noncomputable section
namespace Zarankiewicz
theorem segmentPairTransverse_of_noncollinear_endpoints (a b c d : Plane)
(ha : orient2 c d a ≠ 0) (hb : orient2 c d b ≠ 0)
(hc : orient2 a b c ≠ 0) (hd : orient2 a b d ≠ 0) :
SegmentPairTransverse a b c d := by
intro t ht u hu he
have hdet : det2 (b-a) (d-c) ≠ 0 := by
intro hz
have hdiff := orient2_difference a b c d
have heq : orient2 a b c=orient2 a b d := by linarith
have hh := congrArg (orient2 a b) he
rw [orient2_self_segment,orient2_segment,← heq] at hh
apply hc
nlinarith
have ht0 : t ≠ 0 := by
intro h
subst t
rw [segmentPoint_zero] at he
exact ha (by rw [he,orient2_self_segment])
have ht1 : t ≠ 1 := by
intro h
subst t
rw [segmentPoint_one] at he
exact hb (by rw [he,orient2_self_segment])
have hu0 : u ≠ 0 := by
intro h
subst u
rw [segmentPoint_zero] at he
exact hc (by rw [← he,orient2_self_segment])
have hu1 : u ≠ 1 := by
intro h
subst u
rw [segmentPoint_one] at he
exact hd (by rw [← he,orient2_self_segment])
exact ⟨hdet,⟨lt_of_le_of_ne ht.1 ht0.symm,lt_of_le_of_ne ht.2 ht1⟩,
⟨lt_of_le_of_ne hu.1 hu0.symm,lt_of_le_of_ne hu.2 hu1⟩⟩
theorem circle_orient_ne_zero (a b c : ℝ) (hab : a ≠ b) (hac : a ≠ c) (hbc : b ≠ c) :
orient2 (circlePoint a) (circlePoint b) (circlePoint c) ≠ 0 := by
rw [circle_orient_formula]
apply div_ne_zero
· exact mul_ne_zero (mul_ne_zero (mul_ne_zero (by norm_num) (sub_ne_zero.mpr hab.symm))
(sub_ne_zero.mpr hac.symm)) (sub_ne_zero.mpr hbc.symm)
· exact mul_ne_zero (mul_ne_zero (ne_of_gt (circle_denominator_pos a))
(ne_of_gt (circle_denominator_pos b))) (ne_of_gt (circle_denominator_pos c))
theorem circle_chords_transverse (a b c d : ℝ)
(hab : a ≠ b) (hcd : c ≠ d) (hac : a ≠ c) (had : a ≠ d) (hbc : b ≠ c) (hbd : b ≠ d) :
SegmentPairTransverse (circlePoint a) (circlePoint b) (circlePoint c) (circlePoint d) := by
exact segmentPairTransverse_of_noncollinear_endpoints _ _ _ _
(circle_orient_ne_zero c d a hcd hac.symm had.symm)
(circle_orient_ne_zero c d b hcd hbc.symm hbd.symm)
(circle_orient_ne_zero a b c hab hac hbc)
(circle_orient_ne_zero a b d hab had hbd)
end Zarankiewicz
end
end OAI
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