Question 55
Let $H$ be the orthocenter of an acute triangle $ABC$, and let $D$ be the intersection of the two lines $AH$, $BC$. Let $E$ be the point of intersection of the circumcircle to the triangle $ABD$ and the line $CH$, lying outside of the triangle $ABC$. And let $F$ be the point of intersection of the circumcircle to the triangle $ACD$ and the line $BH$, lying outside of the triangle $ABC$. Show that the two line segments $AE$ and $AF$ have the same length.