/-
Copyright (c) 2021 Manuel Candales. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Manuel Candales, Benjamin Davidson
-/
module

public import Mathlib.Geometry.Euclidean.Sphere.Power
public import Mathlib.Geometry.Euclidean.Triangle

/-!
# Ptolemy's theorem

This file proves Ptolemy's theorem on the lengths of the diagonals and sides of a cyclic
quadrilateral.

## Main theorems

* `mul_dist_add_mul_dist_eq_mul_dist_of_cospherical`: Ptolemy’s Theorem (Freek No. 95).

TODO: The current statement of Ptolemy’s theorem works around the lack of a "cyclic polygon" concept
in mathlib, which is what the theorem statement would naturally use (or two such concepts, since
both a strict version, where all vertices must be distinct, and a weak version, where consecutive
vertices may be equal, would be useful; Ptolemy's theorem should then use the weak one).

An API needs to be built around that concept, which would include:
- strict cyclic implies weak cyclic,
- weak cyclic and consecutive points distinct implies strict cyclic,
- weak/strict cyclic implies weak/strict cyclic for any subsequence,
- any three points on a sphere are weakly or strictly cyclic according to whether they are distinct,
- any number of points on a sphere intersected with a two-dimensional affine subspace are cyclic in
  some order,
- a list of points is cyclic if and only if its reversal is,
- a list of points is cyclic if and only if any cyclic permutation is, while other permutations
  are not when the points are distinct,
- a point P where the diagonals of a cyclic polygon cross exists (and is unique) with weak/strict
  betweenness depending on weak/strict cyclicity,
- four points on a sphere with such a point P are cyclic in the appropriate order,

and so on.
-/

public section


open Real

open scoped EuclideanGeometry RealInnerProductSpace Real

namespace EuclideanGeometry

variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V]
variable {P : Type*} [MetricSpace P] [NormedAddTorsor V P]

/-- **Ptolemy’s Theorem**. -/
theorem mul_dist_add_mul_dist_eq_mul_dist_of_cospherical {a b c d p : P}
    (h : Cospherical ({a, b, c, d} : Set P)) (hapc : ∠ a p c = π) (hbpd : ∠ b p d = π) :
    dist a b * dist c d + dist b c * dist d a = dist a c * dist b d := by sorry
end EuclideanGeometry