Theorems · Theorem · geometry

Affine.Simplex.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] {n : ℕ}
  (s : Affine.Simplex 𝕜 P n) {p₁ : P} (p₂ : P),
  p₁ ∈ affineSpan 𝕜 (Set.range s.points) →
    dist p₁ p₂ * dist p₁ p₂ =
      dist p₁ ↑(s.orthogonalProjectionSpan p₂) * dist p₁ ↑(s.orthogonalProjectionSpan p₂) +
        dist p₂ ↑(s.orthogonalProjectionSpan p₂) * dist p₂ ↑(s.orthogonalProjectionSpan p₂)

Defined in

Mathlib.Geometry.Euclidean.Projection

Cited by

1 results in Mathlib

Foundations

Depth 182 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceMetricSpaceNormedAddTorsor