Theorems · Theorem · geometry

EuclideanGeometry.Sphere.isTangentAt_iff_dist_sq_eq_power

∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] {P : Type u_2} [inst_2 : MetricSpace P]
  [inst_3 : NormedAddTorsor V P] {t p : P} {s : EuclideanGeometry.Sphere P},
  t ∈ s → (s.IsTangentAt t line[ℝ, p, t] ↔ dist p t ^ 2 = s.power p)

A line through a point on a sphere is tangent if and only if the squared distance from the external point to the tangent point equals the power of the point.

Defined in

Mathlib.Geometry.Euclidean.Sphere.Power

Cited by

1 results in Mathlib

Foundations

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

Assumes

NormedAddCommGroupInnerProductSpaceMetricSpaceNormedAddTorsor