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.)