Theorems · Theorem · convex and discrete geometry

Collinear.wbtw_of_dist_eq_of_dist_le

∀ {V : Type u_1} {P : Type u_2} [inst : NormedAddCommGroup V] [inst_1 : NormedSpace ℝ V] [StrictConvexSpace ℝ V]
  [inst_3 : PseudoMetricSpace P] [inst_4 : NormedAddTorsor V P] {p p₁ p₂ p₃ : P} {r : ℝ},
  Collinear ℝ {p₁, p₂, p₃} → dist p₁ p = r → dist p₂ p ≤ r → dist p₃ p = r → p₁ ≠ p₃ → Wbtw ℝ p₁ p₂ p₃

Given three collinear points, two (not equal) with distance r from p and one with distance at most r from p, the third point is weakly between the other two points.

Defined in

Mathlib.Analysis.Convex.StrictConvexBetween

Cited by

2 results in Mathlib

Foundations

Depth 167 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceStrictConvexSpacePseudoMetricSpaceNormedAddTorsor