Theorems · Theorem · geometry

EuclideanGeometry.law_cos

∀ {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] (p₁ p₂ p₃ : P),
  dist p₁ p₃ * dist p₁ p₃ =
    dist p₁ p₂ * dist p₁ p₂ + dist p₃ p₂ * dist p₃ p₂ -
      2 * dist p₁ p₂ * dist p₃ p₂ * Real.cos (EuclideanGeometry.angle p₁ p₂ p₃)

Alias of EuclideanGeometry.dist_sq_eq_dist_sq_add_dist_sq_sub_two_mul_dist_mul_dist_mul_cos_angle. Law of cosines (cosine rule), angle-at-point form.

Defined in

Mathlib.Geometry.Euclidean.Triangle

Cited by

5 results in Mathlib

Foundations

Depth 194 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupInnerProductSpaceMetricSpaceNormedAddTorsor