VN-TST 2019 Problem 3

Given an acute scalene triangle $ABC$ inscribed in circle $(O)$ . Let $H$ be its orthocenter and $M$ be the midpoint of $BC$ . Let $D$ lie on the opposite rays of $HA$ so that $BC=2DM$ . Let $D'$ be the reflection of $D$ through line $BC$ and $X$ be the intersection of $AO$ and $MD$ .

a) Show that $AM$ bisects $D'X$ .

b) Similarly, we define the points $E,F$ like $D$ and $Y,Z$ like $X$ . Let $S$ be the intersection of tangent lines from $B,C$ with respect to $(O)$ . Let $G$ be the projection of the midpoint of $AS$ to the line $AO$ . Show that there exists a point with the same power to all the circles $(BEY),(CFZ),(SGO)$ and $(O)$ .