Theorems · Theorem · geometry

EuclideanGeometry.collinear_iff_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)]
  {p₁ p₂ p₃ p₄ p₅ p₆ : P},
  2 • EuclideanGeometry.oangle p₁ p₂ p₃ = 2 • EuclideanGeometry.oangle p₄ p₅ p₆ →
    (Collinear ℝ {p₁, p₂, p₃} ↔ Collinear ℝ {p₄, p₅, p₆})

If twice the oriented angles between two triples of points are equal, one triple is collinear if and only if the other is.

Defined in

Mathlib.Geometry.Euclidean.Angle.Oriented.Affine

Cited by

3 results in Mathlib

Foundations

Depth 284 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupInnerProductSpaceMetricSpaceNormedAddTorsorFactModule.Oriented