Question 136
Points $C_1$ and $B_1$ are marked on the sides $AB$ and $AC$ of the triangle $ABC$ respectively. Segments $BB_1$ and $CC_1$ intersect at point $X$, and segments $B_1C_1$ and $AX$ intersect at point $A_1$. The circumcircles of the triangles $BXC_1$ and $CXB_1$ intersect the side $BC$ at points $D$ and $E$ respectively. Lines $B_1D$ and $C_1E$ intersect at point $F$.
Prove that the lines $A_1F$, $B_1E$ and $C_1D$ are either parallel or concurrent.
Prove that the lines $A_1F$, $B_1E$ and $C_1D$ are either parallel or concurrent.