KöMaL 2005 B3865

The triangles $AB_1 C_1$ and $AB_2 C_2$ are similar, and their vertices are labelled in the same direction around the triangle. The lines $B_1 B_2$ and $C_1 C_2$ intersect at the point $D$. ($a$) Show that the triangles $AB_1 B_2$ and $AC_1 C_2$ are similar. ($b$) Show that the circumscribed circles of $AB_1 C_1$ and $AB_2 C_2$ pass through the point $D$.