Ptolemy's Theorem
This theorem was proved by Giovanni Ceva (1648-1734).
Ptolemy's theorem states that given a cyclic quadrilateral (i.e. one that can be inscribed in a circle) the product of the diagonals equals the sum of the products of opposite sides.
On the diagonal BD locate a point M such that angles BCA and MCD are equal. Since angles BAC and MDC subtend the same arc, they are equal. (why?) Therefore, triangles ABC and DMC are similar.
Thus we get CD/MD = AC/AB, or AB·CD = AC·MD.
Since angles BCA and MCD are equal, then angle BCM=BCA+ACM equals angle ACD=ACM+MCD. So triangles BCM and ACD are similar which leads to
BC/BM = AC/AD, or BC·AD = AC·BM. …
