Question 186
The circles $\Gamma_1$, $\Gamma_2$, and $\Gamma_3$ in the plane are pairwise externally tangent. Let $P_2$ be the point of tangency between the circles $\Gamma_1$ and $\Gamma_3$, and $P_1$ the point of tangency between the circles $\Gamma_2$ and $\Gamma_3$. Consider points $A$ and $B$ on the circle $\Gamma_3$ that are diametrically opposite, such that the quadrilateral $ABP_1P_2$ is convex.
The line through $A$ and $P_2$ intersects the circle $\Gamma_1$ a second time at point $X$, the line through $B$ and $P_1$ intersects the circle $\Gamma_2$ a second time at point $Y$, and the lines $AP_1$ and $BP_2$ intersect at $Z$.
Prove that the points $X, Y$, and $Z$ are collinear.
The line through $A$ and $P_2$ intersects the circle $\Gamma_1$ a second time at point $X$, the line through $B$ and $P_1$ intersects the circle $\Gamma_2$ a second time at point $Y$, and the lines $AP_1$ and $BP_2$ intersect at $Z$.
Prove that the points $X, Y$, and $Z$ are collinear.