Question 4
Let $ABC$ be an acute scalene triangle inscribed in circle $(O)$, with orthocenter $H$. Let $D,E,F$ be the reflections of $O$ across $BC,CA,AB$, respectively.
a) Let $H_a$ be the reflection of $H$ across $BC$, let $A'$ be the reflection of $A$ across $O$, and let $O_a$ be the circumcenter of $BOC$. Prove that $H_aD$ and $O_aA'$ meet on $(O)$.
b) Choose $X$ so that $AXDA'$ is a parallelogram. Prove that the circumcircles of $AHX$, $ABF$, and $ACE$ have a second common point different from $A$.
Editorial note: the printed assertion in (b) has an allowed exceptional configuration; the solution explains it without changing the source statement.
a) Let $H_a$ be the reflection of $H$ across $BC$, let $A'$ be the reflection of $A$ across $O$, and let $O_a$ be the circumcenter of $BOC$. Prove that $H_aD$ and $O_aA'$ meet on $(O)$.
b) Choose $X$ so that $AXDA'$ is a parallelogram. Prove that the circumcircles of $AHX$, $ABF$, and $ACE$ have a second common point different from $A$.
Editorial note: the printed assertion in (b) has an allowed exceptional configuration; the solution explains it without changing the source statement.