Points $A_{1}, B_{1}, C_{1}$ are the midpoints of the sides $B C, A C, A B$ of triangle $A B C$, respectively. It is known that $A_{1} A$ and $B_{1} B$ are the angle bisectors of triangle $A_{1} B_{1} C_{1}$. Find the angles of triangle $A B C$.
As is known, $A_{1} B_{1} \| A B$ and $A_{1} C_{1} \| A C$.

First method. By the property of parallel lines, $\angle B A A_{1}=\angle A A_{1} B_{1}=\angle A A_{1} C_{1}=\angle A_{1} A C$, hence, the median $A A_{1}$ of triangle $A B C$ is its bisector, so $A B=A C$. Similarly, $A B=B C$. Therefore, triangle $A B C$ is equilateral.

Second method. Quadrilateral $A B_{1} A_{1} C_{1}$ is a parallelogram, in which the diagonal is the bisector of the angle, hence, $A B_{1} A_{1} C_{1}$ is a rhombus. Therefore, $A_{1} B_{1}=A_{1} C_{1}$. Similarly, $A_{1} B_{1}=B_{1} C_{1}$. Thus, triangle $A_{1} B_{1} C_{1}$ is equilateral. Therefore, triangle $A B C$ is also equilateral.

## Answer

$60^{\circ}, 60^{\circ}, 60^{\circ}$.