Ptolemy's theorem #

THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4.

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.