The four vertices of quadrilateral ABCD, named in order, lie on one circle, and the sides measure AB = 3 cm, BC = 4 cm, CD = 5 cm and DA = 6 cm. What is the value of the product AC × BD?

Ptolemy's theorem says that in a cyclic quadrilateral the product of the diagonals equals the sum of the products of the two pairs of OPPOSITE sides: AC × BD = AB × CD + BC × DA. The opposite pairs here are (AB, CD) and (BC, DA), so AC × BD = $3\times 5+4\times 6=15+24=39{\text{cm}}^2$. The trap 42 pairs up ADJACENT sides, $3\times 4+5\times 6$, and 38 uses the third pairing, $3\times 6+4\times 5$ — only the opposite pairing is the theorem. The value 18 is just the perimeter. (Such a quadrilateral does exist: each side is shorter than the other three together.)