Theorems · Theorem · geometry

EuclideanGeometry.angle_eq_pi_sub_angle_div_two_of_oangle_eq_of_sOppSide

∀ {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)]
  {p₁ p₂ p₃ p₄ : P},
  p₁ ≠ p₂ →
    EuclideanGeometry.oangle p₁ p₂ p₃ = EuclideanGeometry.oangle p₃ p₂ p₄ →
      line[ℝ, p₁, p₂].SOppSide p₃ p₄ → EuclideanGeometry.angle p₁ p₂ p₃ = Real.pi - EuclideanGeometry.angle p₁ p₂ p₄ / 2

If p₃ bisects the angle ∡ p₁ p₂ p₄, and p₃ and p₄ lie on opposite sides of the line p₁ p₂, then the unoriented angle ∠ p₁ p₂ p₃ is π minus half ∠ p₁ p₂ p₄.

Defined in

Mathlib.Geometry.Euclidean.Angle.Oriented.Affine

Cited by

1 results in Mathlib

Foundations

Depth 288 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupInnerProductSpaceMetricSpaceNormedAddTorsorFactModule.Oriented