KöMaL 2026 A939

In the acute and scalene triangle $ABC$ let $A_1$, $B_1$ and $C_1$ denote the feet of altitudes from $A$, $B$ and $C$, respectively. Let $A_2$, $B_2$ and $C_2$ denote the intersection of the internal angle bisectors from $A$, $B$ and $C$, respectively, with the opposite sides. Let $X_1$ denote the point of intersection of lines $A_1B_1$ and $A_2C_2$, and let $X_2$ denote the point of intersection of lines $A_1C_1$ and $A_2B_2$. Similarly, let $Y_1$ denote the point of intersection of lines $B_1C_1$ and $B_2A_2$ and $Y_2$ denote the point of intersection of lines $B_1A_1$ and $B_2C_2$, let $Z_1$ denote the point of intersection of lines $C_1A_1$ and $C_2B_2$, and finally let $Z_2$ denote the point of intersection of lines $C_1B_1$ and $C_2A_2$. Assume that all these points exist and are distinct. Prove that points $A$, $X_1$, $X_2$, points $B$, $Y_1$, $Y_2$ and points $C$, $Z_1$ and $Z_2$ are collinear, and these three lines are concurrent.