Question 51
Points $D,E$ lie on sides $AB,AC$ of triangle $ABC$, respectively, with $DE\parallel BC$. Let $P$ be inside triangle $ADE$, and let $F,G$ be the intersections of $PB,PC$ with segment $DE$, respectively. If $O_1,O_2$ are the circumcentres of triangles $DPG,FPE$, prove that $O_1O_2\perp AP$.