Menu Close

Let-ABCD-be-a-convex-quadrilateral-Prove-that-AB-CD-AD-BC-AC-BD-if-and-only-if-ABCD-is-cyclic-Ptolemy-s-theorem-




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

Leave a Reply

Your email address will not be published. Required fields are marked *