CMO 2013 Problem 1

Two circles $K_1$ and $K_2$ of different radii intersect at two points $A$ and $B$ , let $C$ and $D$ be two points on $K_1$ and $K_2$ , respectively, such that $A$ is the midpoint of the segment $CD$ . The extension of $DB$ meets $K_1$ at another point $E$ , the extension of $CB$ meets $K_2$ at another point $F$ . Let $l_1$ and $l_2$ be the perpendicular bisectors of $CD$ and $EF$ , respectively. i) Show that $l_1$ and $l_2$ have a unique common point (denoted by $P$ ). ii) Prove that the lengths of $CA$ , $AP$ and $PE$ are the side lengths of a right triangle.

Attachments: