Theorems · Theorem · geometry

AffineSubspace.mem_perpBisector_iff_inner_eq_zero

∀ {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] {c p₁ p₂ : P},
  c ∈ AffineSubspace.perpBisector p₁ p₂ ↔ inner ℝ (c -ᵥ midpoint ℝ p₁ p₂) (p₂ -ᵥ p₁) = 0

A point c belongs the perpendicular bisector of [p₁, p₂] iff c -ᵥ midpoint ℝ p₁ p₂ is orthogonal to p₂ -ᵥ p₁.

Defined in

Mathlib.Geometry.Euclidean.PerpBisector

Cited by

6 results in Mathlib

Foundations

Depth 164 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupInnerProductSpaceMetricSpaceNormedAddTorsor