CN-TST 2011 Problem 10

Let $AA',BB',CC'$ be three diameters of the circumcircle of an acute triangle $ABC$ . Let $P$ be an arbitrary point in the interior of $\triangle ABC$ , and let $D,E,F$ be the orthogonal projection of $P$ on $BC,CA,AB$ , respectively. Let $X$ be the point such that $D$ is the midpoint of $A'X$ , let $Y$ be the point such that $E$ is the midpoint of $B'Y$ , and similarly let $Z$ be the point such that $F$ is the midpoint of $C'Z$ . Prove that triangle $XYZ$ is similar to triangle $ABC$ .