Question 32
Let $ABC$ be an acute scalene triangle with circumcenter $O$. Let $D$ be the foot of the altitude from $A$ to the side $BC$. The lines $BC$ and $AO$ intersect at $E$. Let $s$ be the line through $E$ perpendicular to $AO$. The line $s$ intersects $AB$ and $AC$ at $K$ and $L$, respectively. Denote by $\omega$ the circumcircle of triangle $AKL$. Line $AD$ intersects $\omega$ again at $X$.
Prove that $\omega$ and the circumcircles of triangles $ABC$ and $DEX$ have a common point.
Prove that $\omega$ and the circumcircles of triangles $ABC$ and $DEX$ have a common point.