Let $B'$ and $C'$ be points in the circumcircle of triangle $\triangle ABC$ such that $AB=AB'$ and $AC=AC'$. Let $E$ and $F$ be the foot of altitudes from $B$ and $C$ to $AC$ and $AB$, respectively. Show that $B'E$ and $C'F$ intersect on the circumcircle of triangle $\triangle ABC$.