Theorems · Theorem · geometry

Affine.Simplex.orthogonalProjection_eq_circumcenter_of_exists_dist_eq

∀ {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] {n : ℕ} (s : Affine.Simplex ℝ P n) {p : P},
  (∃ r, ∀ (i : Fin (n + 1)), dist (s.points i) p = r) → ↑(s.orthogonalProjectionSpan p) = s.circumcenter

If there exists a distance that a point has from all vertices of a simplex, the orthogonal projection of that point onto the subspace spanned by that simplex is its circumcenter.

Defined in

Mathlib.Geometry.Euclidean.Circumcenter

Cited by

2 results in Mathlib

Foundations

Depth 191 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupInnerProductSpaceMetricSpaceNormedAddTorsor