# Power of a point (intersecting chords and secants) #

This file proves basic geometrical results about power of a point (intersecting chords and secants) in spheres in real inner product spaces and Euclidean affine spaces.

## Main theorems #

• mul_dist_eq_mul_dist_of_cospherical_of_angle_eq_pi: Intersecting Chords Theorem (Freek No. 55).
• mul_dist_eq_mul_dist_of_cospherical_of_angle_eq_zero: Intersecting Secants Theorem.

### Geometrical results on spheres in real inner product spaces #

This section develops some results on spheres in real inner product spaces, which are used to deduce corresponding results for Euclidean affine spaces.

theorem InnerProductGeometry.mul_norm_eq_abs_sub_sq_norm {V : Type u_1} [] {x : V} {y : V} {z : V} (h₁ : ∃ (k : ), k 1 x + y = k (x - y)) (h₂ : z - y = z + y) :
x - y * x + y = |z + y ^ 2 - z - x ^ 2|

### Geometrical results on spheres in Euclidean affine spaces #

This section develops some results on spheres in Euclidean affine spaces.

theorem EuclideanGeometry.mul_dist_eq_abs_sub_sq_dist {V : Type u_1} [] {P : Type u_2} [] [] {a : P} {b : P} {p : P} {q : P} (hp : ∃ (k : ), k 1 b -ᵥ p = k (a -ᵥ p)) (hq : 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).

theorem EuclideanGeometry.mul_dist_eq_mul_dist_of_cospherical {V : Type u_1} [] {P : Type u_2} [] [] {a : P} {b : P} {c : P} {d : P} {p : P} (h : EuclideanGeometry.Cospherical {a, b, c, d}) (hapb : ∃ (k₁ : ), k₁ 1 b -ᵥ p = k₁ (a -ᵥ p)) (hcpd : ∃ (k₂ : ), k₂ 1 d -ᵥ p = k₂ (c -ᵥ p)) :
dist a p * dist b p = dist c p * dist d p

If A, B, C, D are cospherical and P is on both lines AB and CD, then AP * BP = CP * DP.

theorem EuclideanGeometry.mul_dist_eq_mul_dist_of_cospherical_of_angle_eq_pi {V : Type u_1} [] {P : Type u_2} [] [] {a : P} {b : P} {c : P} {d : P} {p : P} (h : EuclideanGeometry.Cospherical {a, b, c, d}) (hapb : ) (hcpd : ) :
dist a p * dist b p = dist c p * dist d p

Intersecting Chords Theorem.

theorem EuclideanGeometry.mul_dist_eq_mul_dist_of_cospherical_of_angle_eq_zero {V : Type u_1} [] {P : Type u_2} [] [] {a : P} {b : P} {c : P} {d : P} {p : P} (h : EuclideanGeometry.Cospherical {a, b, c, d}) (hab : a b) (hcd : c d) (hapb : = 0) (hcpd : = 0) :
dist a p * dist b p = dist c p * dist d p

Intersecting Secants Theorem.