Question 110
Let $H$ be the orthocenter of the acute triangle $ABC$ and let $X$ be the midpoint of the side $BC$. The perpendicular at $H$ to $HX$ intersects the sides $AB$ and $AC$ at the points $Y$ and $Z$ respectively. Let $O$ be the circumcenter of the triangle $ABC$ and $O'$ be the circumcenter of the triangle $BHC$. Prove that:
a) $HY = HZ$;
b) $\overrightarrow{AY} + \overrightarrow{AZ} = 2\overrightarrow{OO'}$.
a) $HY = HZ$;
b) $\overrightarrow{AY} + \overrightarrow{AZ} = 2\overrightarrow{OO'}$.