Theorems · Theorem · geometry

Affine.Simplex.excenterWeights_empty_lt_inv_two

∀ {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} [inst_4 : NeZero n] (s : Affine.Simplex ℝ P n) [n.AtLeastTwo]
  (i : Fin (n + 1)), s.excenterWeights ∅ i < 2⁻¹

The barycentric coordinates of the incenter are less than 2⁻¹ (thus, it lies closer on an angle bisector to the opposite side than to the vertex, or equivalently the image of the incenter under a homothety with scale factor 2 about a vertex lies outside the simplex).

Defined in

Mathlib.Geometry.Euclidean.Incenter

Cited by

0 results in Mathlib

Foundations

Depth 190 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupInnerProductSpaceMetricSpaceNormedAddTorsorNeZeroNat.AtLeastTwo