Abstract
lABI * ICDI + IBCI * IDAI = IACI IBDI, (1.1) where IABI denotes Euclidean length. 'Ptolemy's theorem' (we do not know whether he actually discovered it) proves to be so useful because it is equivalent to the trigonometric addition formulae. If BD is a diameter of length 2 R and angle ADB = e, angle BDC = 7q then sin e cos -q + cos ( sin -q = sin(e + 7q) follows easily. Ptolemy's methods are described in [1] and [3]; his tables were used in astronomical calculations for a thousand years. We call a quadrilateral 'cyclic' if its vertices lie in the given order on a circle. When the quadrilateral ABCD is not cyclic the sides and diagonals satisfy a strict inequality
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