Question 112
The square $A_1B_1C_1D_1$ is inscribed in the right triangle $ABC$ (with $\angle C = 90^\circ$) so that the points $A_1, B_1$ lie on the legs $CB$ and $CA$ respectively, and the points $C_1, D_1$ lie on the hypotenuse $AB$. The circumcircles of the triangles $B_1A_1C$ and $BD_1A_1$ intersect at $A_1$ and $X$, and the circumcircles of the triangles $B_1A_1C$ and $AC_1B_1$ intersect at $B_1$ and $Y$.
Prove that the lines $A_1X$ and $B_1Y$ meet on the hypotenuse $AB$.
Prove that the lines $A_1X$ and $B_1Y$ meet on the hypotenuse $AB$.