Question 193
Let $ABC$ be a triangle. Points $E, F$ move on the rays opposite to $BA$ and $CA$, respectively, such that $BF = CE$. Let $M, N$ be the midpoints of $BE, CF$. Suppose that $BF$ cuts $CE$ at $D$.
a) Let $I, J$ be the centers of the circumcircles of triangles $DBE$ and $DCF$. Prove that $MN$ is parallel to $IJ$.
b) Let $K$ be the midpoint of $MN$ and $H$ be the orthocenter of triangle $AEF$. Prove that when $E$ moves on the ray opposite to $BA$, the line $HK$ goes through a fixed point.
a) Let $I, J$ be the centers of the circumcircles of triangles $DBE$ and $DCF$. Prove that $MN$ is parallel to $IJ$.
b) Let $K$ be the midpoint of $MN$ and $H$ be the orthocenter of triangle $AEF$. Prove that when $E$ moves on the ray opposite to $BA$, the line $HK$ goes through a fixed point.