Question 3
Let $ABC$ be acute. Points $E,F$ vary on the rays opposite to $BA,CA$, respectively, with $BF=CE$, $E\ne B$, $F\ne C$. Let $M,N$ be the midpoints of $BE,CF$, and $D=BF\cap CE$.
(a) Let $I,J$ be the circumcenters of $DBE,DCF$, respectively. Prove $MN\parallel IJ$.
(b) Let $K$ be the midpoint of $MN$, and let $H$ be the orthocenter of $AEF$. Prove that $HK$ passes through a fixed point.
(a) Let $I,J$ be the circumcenters of $DBE,DCF$, respectively. Prove $MN\parallel IJ$.
(b) Let $K$ be the midpoint of $MN$, and let $H$ be the orthocenter of $AEF$. Prove that $HK$ passes through a fixed point.