Theorems · Theorem · geometry

EuclideanGeometry.dist_sq_eq_dist_orthogonalProjection_sq_add_dist_orthogonalProjection_sq

∀ {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup V]
  [inst_2 : InnerProductSpace 𝕜 V] [inst_3 : MetricSpace P] [inst_4 : NormedAddTorsor V P] {s : AffineSubspace 𝕜 P}
  [inst_5 : Nonempty ↥s] [inst_6 : s.direction.HasOrthogonalProjection] {p₁ : P} (p₂ : P),
  p₁ ∈ s →
    dist p₁ p₂ * dist p₁ p₂ =
      dist p₁ ↑((EuclideanGeometry.orthogonalProjection s) p₂) *
          dist p₁ ↑((EuclideanGeometry.orthogonalProjection s) p₂) +
        dist p₂ ↑((EuclideanGeometry.orthogonalProjection s) p₂) *
          dist p₂ ↑((EuclideanGeometry.orthogonalProjection s) p₂)

The square of the distance from a point in s to p₂ equals the sum of the squares of the distances of the two points to the orthogonalProjection.

Defined in

Mathlib.Geometry.Euclidean.Projection

Cited by

5 results in Mathlib

Foundations

Depth 181 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceMetricSpaceNormedAddTorsorNonemptySubmodule.HasOrthogonalProjection