Question 24
Let $ABC$ be a triangle with circumcenter $O$. Points $D,E,F$ lie respectively on sides $BC,CA,AB$, such that line $DE$ is perpendicular to $CO$ and line $DF$ is perpendicular to $BO$. (For example, $D$ lies on line $BC$ and is between $B$ and $C$ on that line.)
Let $K$ be the circumcenter of triangle $AFE$. Prove that $DK$ and $BC$ are perpendicular.
Let $K$ be the circumcenter of triangle $AFE$. Prove that $DK$ and $BC$ are perpendicular.