Theorems · Theorem · geometry

EuclideanGeometry.Sphere.vadd_mem_inter_orthRadius_iff_norm_sq

∀ {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} {p : P} {v : V},
  0 ≤ s.radius →
    v ∈ (ℝ ∙ (p -ᵥ s.center))ᗮ →
      (v +ᵥ p ∈ Metric.sphere s.center s.radius ∩ ↑(s.orthRadius p) ↔ ‖v‖ ^ 2 = s.radius ^ 2 - dist p s.center ^ 2)

Defined in

Mathlib.Geometry.Euclidean.Sphere.OrthRadius

Cited by

2 results in Mathlib

Foundations

Depth 171 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupInnerProductSpaceMetricSpaceNormedAddTorsor