Question 59
$A$, $B$ and $C$ are points on the circumference of a circle with centre $O$, such that $\triangle ABC$ is not a right-angled triangle. The point $P$ lies on the circumcircle $\Gamma_1$ of the triangle $OAB$ such that $OP$ is a diameter of $\Gamma_1$. The point $Q$ lies on the circumcircle $\Gamma_2$ of the triangle $OAC$ such that $OQ$ is a diameter of $\Gamma_2$. Tangents are drawn to the circles $\Gamma_1$ and $\Gamma_2$ at $P$ and $Q$ respectively; these two tangents intersect at $K$. The line $CA$ meets the circle $\Gamma_1$ at $A$ and $X$. Prove that $X$ lies on the line $KO$.