Question 122
The altitudes $CC_1$ and $BB_1$ are drawn in the acute triangle $ABC$. The bisectors of the angles $\angle BB_1C$ and $\angle CC_1B$ intersect the line $BC$ at the points $D$ and $E$ respectively and meet each other at the point $X$.
Prove that the intersection points of the circumcircles of the triangles $BEX$ and $CDX$ lie on the line $AX$.
Prove that the intersection points of the circumcircles of the triangles $BEX$ and $CDX$ lie on the line $AX$.