Theorems · Theorem · geometry

EuclideanGeometry.dist_smul_vadd_eq_dist

∀ {V : Type u_1} {P : Type u_2} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] [inst_2 : MetricSpace P]
  [inst_3 : NormedAddTorsor V P] {v : V} (p₁ p₂ : P),
  v ≠ 0 → ∀ (r : ℝ), dist (r • v +ᵥ p₁) p₂ = dist p₁ p₂ ↔ r = 0 ∨ r = -2 * inner ℝ v (p₁ -ᵥ p₂) / inner ℝ v v

The condition for two points on a line to be equidistant from another point.

Defined in

Mathlib.Geometry.Euclidean.Basic

Cited by

4 results in Mathlib

Foundations

Depth 171 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupInnerProductSpaceMetricSpaceNormedAddTorsor