Theorems · Theorem · geometry

Affine.Triangle.eq_excenter_of_two_zsmul_oangle_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] [hd2 : Fact (Module.finrank ℝ V = 2)] [inst_4 : Module.Oriented ℝ V (Fin 2)]
  {t : Affine.Triangle ℝ P} {i₁ i₂ i₃ : Fin 3},
  i₁ ≠ i₂ →
    i₁ ≠ i₃ →
      i₂ ≠ i₃ →
        ∀ {p : P},
          2 • EuclideanGeometry.oangle (t.points i₂) (t.points i₁) p =
              2 • EuclideanGeometry.oangle p (t.points i₁) (t.points i₃) →
            2 • EuclideanGeometry.oangle (t.points i₃) (t.points i₂) p =
                2 • EuclideanGeometry.oangle p (t.points i₂) (t.points i₁) →
              ∃ signs, p = Affine.Simplex.excenter t signs

A point lying on angle bisectors from two vertices is an excenter.

Defined in

Mathlib.Geometry.Euclidean.Angle.Incenter

Cited by

3 results in Mathlib

Foundations

Depth 293 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupInnerProductSpaceMetricSpaceNormedAddTorsorFactModule.Oriented