Question 19
Let $ABC$ be an acute-angled triangle with $AB \lt AC$. Let $\Gamma$ be the circumcircle of triangle $ABC$. The perpendicular from $B$ to $AC$ intersects $\Gamma$ again at $D \neq B$. Let $Q$ be the point on $\Gamma$ such that $Q \neq B$ and $AB = AQ$. Lines $BQ$ and $AC$ intersect at $R$. Line $DR$ intersects $\Gamma$ again at $S \neq D$. Segment $BC$ intersects the circumcircle of triangle $QRS$ at $T$. Lines $TQ$ and $RS$ intersect at $X$.
Prove that CX is perpendicular to AB.
Prove that CX is perpendicular to AB.