Question 185
Let $ABC$ be a non-isosceles acute triangle and let $\ell$ be the bisector of angle $A$. A circle with center $O$ passes through $B, C$ and intersects the sides $AB, AC$ at $D, E$ respectively. Let $T$ be the intersection of $BE$ and $CD$, and construct a parallelogram $DTEK$. Take $X, Y$ on $AB, AC$ respectively such that $AXKY$ is also a parallelogram. Construct $P, Q$ on $XY$ such that $TP \parallel AC$ and $TQ \parallel AB$. Prove that $TO$ passes through the center of the circumcircle of triangle $TPQ$.