Ptolemy's theorem and extensions

Ptolemy's theorem is where serious trigonometry starts, although it is chords of circles rather than right triangles which are the building blocks. Ptolemy's theorem deals with cyclic quadrilaterals, their sides and the diagonals. In Ptolemy's book, both the statement and the proof seem to come out of nowhere. I will start with the motivation for the theorem, give Ptolemy’s beautiful proof and then give some other proofs and an extension to a general quadrilateral. The proof for a general quadrilateral which I will give combines one idea from Ptolemy's proof and of a proof Euler gave of a different extension of Ptolemy’s theorem. All of this can be understood with a knowledge of high school mathematics, so students are encouraged to attend.