Theorems · Theorem · geometry

EuclideanGeometry.inner_vsub_center_vsub_pos

∀ {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 : P} {s : EuclideanGeometry.Sphere P},
  p₁ ∈ s → p₂ ∈ s → p₁ ≠ p₂ → 0 < inner ℝ (p₂ -ᵥ p₁) (s.center -ᵥ p₁)

Given two distinct points on a sphere, the inner product of the chord with the radius vector at one endpoint is negative.

Defined in

Mathlib.Geometry.Euclidean.Sphere.Basic

Cited by

0 results in Mathlib

Foundations

Depth 163 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupInnerProductSpaceMetricSpaceNormedAddTorsor