Theorems · Theorem · geometry

EuclideanGeometry.Sphere.mul_dist_eq_neg_power_of_dist_center_le_radius

∀ {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] {s : EuclideanGeometry.Sphere P} {p a b : P},
  0 ≤ s.radius → p ∈ line[ℝ, a, b] → a ∈ s → b ∈ s → dist p s.center ≤ s.radius → dist p a * dist p b = -s.power p

For a point inside or on the sphere, the product of distances to two intersection points on a line through the point equals the negative of the power of the point.

Defined in

Mathlib.Geometry.Euclidean.Sphere.Power

Cited by

0 results in Mathlib

Foundations

Depth 205 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupInnerProductSpaceMetricSpaceNormedAddTorsor