Question 149
The points $A$, $B$, $K$, $L$, $X$ lie on the circle $\Gamma$ in the given order, and the arcs $\widehat{BK}$ and $\widehat{KL}$ have equal lengths. The circle which passes through $A$ and is tangent to $BK$ at $B$ intersects the line segment $KX$ at points $P$ and $Q$. The second intersection point of the circle which passes through $A$ and is tangent to $BL$ at $B$ with the line segment $BX$ is $T$. Show that $\angle PTB = \angle XTQ$.