Theorems · Theorem · geometry

Affine.Simplex.point_vsub_centroid_eq_smul_vsub

• 1000+ list: Commandino's theorem

∀ {k : Type u_1} {V : Type u_2} {P : Type u_3} [inst : DivisionRing k] [inst_1 : AddCommGroup V] [inst_2 : Module k V]
  [inst_3 : AddTorsor V P] {n : ℕ} [inst_4 : NeZero n] [CharZero k] (s : Affine.Simplex k P n) (i : Fin (n + 1)),
  s.points i -ᵥ s.centroid = ↑n • (s.centroid -ᵥ s.faceOppositeCentroid i)

Commandino's theorem : For n-simplex, the vector from a vertex to the centroid equals n times the vector from the centroid to the corresponding faceOppositeCentroid.

Defined in

Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid

Cited by

4 results in Mathlib

Foundations

Depth 78 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

DivisionRingAddCommGroupModuleAddTorsorNeZeroCharZero