Circumcircle and excircles

[ins-]

Triangle $ABC$ is inscribed in circle $k$. $k_1, k_2, k_3$ are its excircles tangent to the sides $AB$, $BC$, $CA$ respectively. $k$ and $k_1$ intersects at the points $X_1$ and $X_2$. $k$ and $k_2$ intersects at the points $Y_1$ and $Y_2$. $k$ and $k_3$ intersects at the points $Z_1$ and $Z_2$. $A_1$ is the intersection point of $Z_1Z_2$ and $X_1X_2$. $B_1$ is the intersection point of $X_1X_2$ and $Y_1Y_2$. $C_1$ is the intersection point of $Y_1Y_2$ and $Z_1Z_2$. Prove that $AA_1$, $BB_1$, $CC_1$ intersects at a common point.

[ins-]

If you are interested you can see a solution. I don't like that problem very much but for the sake of completeness I posted it.

(Оффтоп)

http://www.artofproblemsolving.com/Forum/viewtopic.php?f=48&t=481693