Question 74
Three nonintersecting circles $k_{i}(O_{i}, r_{i})$, $i=1,2,3$, where $r_{1}\lt r_{2}\lt r_{3}$, are tangent to the arms of an angle. One of the arms is tangent to $k_{1}$ and $k_{3}$ at points $A$ and $B$ and the other one is tangent to $k_{2}$ at point $C$. Let $K=AC \cap k_{1}$, $L=AC \cap k_{2}$, $M=BC \cap k_{2}$ and $N=BC \cap k_{3}$. The four lines through $C$ and $P=AM \cap BK$, $Q=AM \cap BL$, $R=AN \cap BK$ and $S=AN \cap BL$, meet $AB$ at the points $X, Y, Z$ and $T$, respectively. Prove that $XZ=YT$.