Question 6
Let $O$ and $H$ be respectively the circumcentre and orthocentre of $\triangle ABC$. Let $A'$, $B'$ and $C'$ be the midpoints of $BC$, $CA$ and $AB$ respectively and $D$, $E$ and $F$ be respectively the feet from $A$, $B$ and $C$ to the opposite sides. Show that $OA' \cdot HD = OB' \cdot HE = OC' \cdot HF$.