An alternate proof for a regular pentagon can be given by a
method that is of interest because it makes use of Ptolemy’s
remarkable theorem on cyclic quadrilaterals (quadrilaterals
inscribed in a circle). Ptolemy’s theorem states that, for any
cyclic quadrilateral, the product of the lengths of the diagonals
is equal to the sum of the products of the lengths of
opposite sides. (A simple proof of Ptolemy’s theorem can be
found in the Workbook that accompanies the videotape Sines
& Cosines, Part III, produced by Project MATHEMATICS!,
Caltech, 1994. The videotape also gives a computer animated
version of this proof.)