Theorems · Theorem · geometry

EuclideanGeometry.two_zsmul_oangle_eq_of_dist_orthogonalProjection_line_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] [inst_4 : Fact (Module.finrank ℝ V = 2)] [inst_5 : Module.Oriented ℝ V (Fin 2)]
  {p p₁ p₂ p₃ : P},
  AffineIndependent ℝ ![p₁, p₂, p₃] →
    dist p ↑((EuclideanGeometry.orthogonalProjection line[ℝ, p₁, p₂]) p) =
        dist p ↑((EuclideanGeometry.orthogonalProjection line[ℝ, p₁, p₃]) p) →
      2 • EuclideanGeometry.oangle p₂ p₁ p = 2 • EuclideanGeometry.oangle p p₁ p₃

If a point p is equidistant to two different lines p₁ p₂ and p₁ p₃, the oriented angles at p₁ are equal modulo π.

Defined in

Mathlib.Geometry.Euclidean.Angle.Bisector

Cited by

1 results in Mathlib

Foundations

Depth 289 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupInnerProductSpaceMetricSpaceNormedAddTorsorFactModule.Oriented