Question 139
Two distinct circles $\omega_1$ and $\omega_2$ intersect at points $X$ and $Y$. Let the lines $l_1$ and $l_2$ be the common tangent lines of these circles such that $l_1$ is tangent to $\omega_1$ at $A$ and to $\omega_2$ at $C$, and $l_2$ is tangent to $\omega_1$ at $B$ and to $\omega_2$ at $D$. Let $Z$ be the reflection of $Y$ with respect to $l_1$. Let $BC$ and $\omega_1$ meet at $K$ for the second time. Let $AD$ and $\omega_2$ meet at $L$ for the second time. Prove that the line tangent to $\omega_1$ and passing through $K$ and the line tangent to $\omega_2$ and passing through $L$ meet on the line $XZ$.