Theorems · Theorem · geometry

EuclideanGeometry.angle_eq_iff_oangle_eq_or_wbtw

∀ {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 →
    p₄ ≠ p₂ →
      (EuclideanGeometry.angle p₁ p₂ p₃ = EuclideanGeometry.angle p₃ p₂ p₄ ↔
        EuclideanGeometry.oangle p₁ p₂ p₃ = EuclideanGeometry.oangle p₃ p₂ p₄ ∨ Wbtw ℝ p₂ p₁ p₄ ∨ Wbtw ℝ p₂ p₄ p₁)

The unoriented angles at p₂ between p₁ and p₃, and between p₃ and p₄, are equal if and only if the oriented angles are equal (p₃ lies on the angle bisector) or one of p₁ and p₄ is weakly between p₂ and the other.

Defined in

Mathlib.Geometry.Euclidean.Angle.Oriented.Affine

Cited by

2 results in Mathlib

Foundations

Depth 287 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupInnerProductSpaceMetricSpaceNormedAddTorsorFactModule.Oriented