Theorems · Theorem · geometry

EuclideanGeometry.Sphere.mul_dist_eq_abs_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] {s : EuclideanGeometry.Sphere P} {p a b : P},
  p ∈ line[ℝ, a, b] → a ∈ s → b ∈ s → dist p a * dist p b = |s.power p|

For any point, the product of distances to two intersection points on a line through the point equals the absolute value of the power of the point.

Defined in

Mathlib.Geometry.Euclidean.Sphere.Power

Cited by

2 results in Mathlib

Foundations

Depth 204 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupInnerProductSpaceMetricSpaceNormedAddTorsor