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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 | import OAI.Combinatorics.Crossing.ParallelDirection
import Mathlib.Analysis.Convex.Basic
namespace OAI
noncomputable section
namespace Zarankiewicz
theorem collinear_segment_subset_union (a b c : Plane) (hac : a ≠ c)
(h : det2 (b-a) (c-b)=0) :
segment ℝ a c ⊆ segment ℝ a b ∪ segment ℝ b c := by
have hd : det2 (c-a) (b-a)=0 := by
have he : det2 (c-a) (b-a) = -det2 (b-a) (c-b) := by dsimp [det2]; ring
rw [he,h,neg_zero]
have hp := vector_parallel_of_det_zero (c-a) (b-a) (sub_ne_zero.mpr (Ne.symm hac)) hd
have hb : b ∈ Set.range (AffineMap.lineMap a c : ℝ → Plane) := by
refine ⟨dot2 (c-a) (b-a)/squaredLength (c-a),?_⟩
rw [AffineMap.lineMap_apply]
change (dot2 (c-a) (b-a)/squaredLength (c-a)) • (c-a)+a=b
rw [← hp,sub_add_cancel]
intro x hx
rw [← insert_endpoints_openSegment] at hx
rcases hx with hxa | hxc | hxo
· subst x
exact Or.inl (left_mem_segment ℝ a b)
· subst x
exact Or.inr (right_mem_segment ℝ b c)
· rcases openSegment_subset_union a c hb hxo with hxb | hx'
· subst x
exact Or.inl (right_mem_segment ℝ a b)
· exact hx'.elim (fun hh => Or.inl (openSegment_subset_segment ℝ a b hh))
(fun hh => Or.inr (openSegment_subset_segment ℝ b c hh))
end Zarankiewicz
end
end OAI
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