Theorems · Theorem · geometry

EuclideanGeometry.Sphere.angle_center_eq_pi_iff_isDiameter

∀ {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] {s : EuclideanGeometry.Sphere P} {p₁ p₂ : P},
  p₁ ∈ s → p₂ ∈ s → s.radius ≠ 0 → (EuclideanGeometry.angle p₁ s.center p₂ = Real.pi ↔ s.IsDiameter p₁ p₂)

On a sphere of nonzero radius, the central angle ∠ p₁ s.center p₂ equals π iff p₁ and p₂ are diametrically opposite.

Defined in

Mathlib.Geometry.Euclidean.Angle.Sphere

Cited by

0 results in Mathlib

Foundations

Depth 200 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupInnerProductSpaceMetricSpaceNormedAddTorsor