Question 3
Let $ABC$ be an acute scalene triangle inscribed in $(O)$, with orthocenter $H$ and midpoint $M$ of $BC$. Let $D$ lie on the ray opposite to $HA$ and satisfy $DM=BC/2$. Let $D'$ be the reflection of $D$ in $BC$, and let $X=AO\cap MD$.
(a) Prove that $AM$ passes through the midpoint of $D'X$.
(b) Define $E,F$ cyclically in the same way as $D$, and $Y,Z$ cyclically in the same way as $X$. Let $S$ be the intersection of the tangents to $(O)$ at $B,C$. Let $G$ be the perpendicular projection onto $AO$ of the midpoint of $AS$. Prove that a point has equal powers with respect to all four circles $(SGO),(BYE),(CFZ),(O)$.
(a) Prove that $AM$ passes through the midpoint of $D'X$.
(b) Define $E,F$ cyclically in the same way as $D$, and $Y,Z$ cyclically in the same way as $X$. Let $S$ be the intersection of the tangents to $(O)$ at $B,C$. Let $G$ be the perpendicular projection onto $AO$ of the midpoint of $AS$. Prove that a point has equal powers with respect to all four circles $(SGO),(BYE),(CFZ),(O)$.