Question Number 16070 by Tinkutara last updated on 21/Jun/17
$$\mathrm{Let}\:{ABCD}\:\mathrm{be}\:\mathrm{a}\:\mathrm{convex}\:\mathrm{quadrilateral}. \\ $$$$\mathrm{Prove}\:\mathrm{that} \\ $$$${AB}.{CD}\:+\:{AD}.{BC}\:=\:{AC}.{BD} \\ $$$$\mathrm{if}\:\mathrm{and}\:\mathrm{only}\:\mathrm{if}\:{ABCD}\:\mathrm{is}\:\mathrm{cyclic}\:\left(\mathrm{Ptolemy}'\mathrm{s}\right. \\ $$$$\left.\mathrm{theorem}\right). \\ $$