Theorems · Theorem · geometry

Affine.Simplex.ExcenterExists.angle_excenter_touchpoint_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} [inst_4 : NeZero n] {s : Affine.Simplex ℝ P n} {signs : Finset (Fin (n + 1))},
  s.ExcenterExists signs →
    ∀ {p : P} {i₁ i₂ : Fin (n + 1)},
      p ∈ affineSpan ℝ (Set.range (s.faceOpposite i₁).points) →
        p ∈ affineSpan ℝ (Set.range (s.faceOpposite i₂).points) →
          EuclideanGeometry.angle (s.excenter signs) p (s.touchpoint signs i₁) =
            EuclideanGeometry.angle (s.excenter signs) p (s.touchpoint signs i₂)

An excenter of a simplex bisects the angle at a point shared between two faces, as measured between that excenter and its touchpoints on those faces.

Defined in

Mathlib.Geometry.Euclidean.Angle.Incenter

Cited by

1 results in Mathlib

Foundations

Depth 205 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupInnerProductSpaceMetricSpaceNormedAddTorsorNeZero