Question 155
In triangle $\triangle A B C$, we have marked points $A_{1}$ on side $B C$, $B_{1}$ on side $A C$, and $C_{1}$ on side $A B$ so that $A A_{1}$ is an altitude, $B B_{1}$ is a median, and $C C_{1}$ is an angle bisector. It is known that $\triangle A_{1} B_{1} C_{1}$ is equilateral. Prove that $\triangle A B C$ is equilateral too.
(Note: A median connects a vertex of a triangle with the midpoint of the opposite side. Thus, for median $B B_{1}$ we know that $B_{1}$ is the midpoint of side $A C$ in $\triangle A B C$.)
(Note: A median connects a vertex of a triangle with the midpoint of the opposite side. Thus, for median $B B_{1}$ we know that $B_{1}$ is the midpoint of side $A C$ in $\triangle A B C$.)