Theorems · Theorem · geometry

EuclideanGeometry.mul_dist_eq_abs_sub_sq_dist

∀ {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] {a b p q : P},
  p ∈ line[ℝ, a, b] → dist a q = dist b q → dist a p * dist b p = |dist b q ^ 2 - dist p q ^ 2|

If P is a point on the line AB and Q is equidistant from A and B, then AP * BP = abs (BQ ^ 2 - PQ ^ 2).

Defined in

Mathlib.Geometry.Euclidean.Sphere.Power

Cited by

3 results in Mathlib

Foundations

Depth 203 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupInnerProductSpaceMetricSpaceNormedAddTorsor