Theorems · Theorem · geometry

EuclideanGeometry.oangle_ne_zero_and_ne_pi_iff_affineIndependent

∀ {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 : P},
  EuclideanGeometry.oangle p₁ p₂ p₃ ≠ 0 ∧ EuclideanGeometry.oangle p₁ p₂ p₃ ≠ ↑Real.pi ↔
    AffineIndependent ℝ ![p₁, p₂, p₃]

An oriented angle is not zero or π if and only if the three points are affinely independent.

Defined in

Mathlib.Geometry.Euclidean.Angle.Oriented.Affine

Cited by

4 results in Mathlib

Foundations

Depth 281 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupInnerProductSpaceMetricSpaceNormedAddTorsorFactModule.Oriented