2013 IMO Shortlist G6

Let the excircle of the triangle $A B C$ lying opposite to $A$ touch its side $B C$ at the point $A_{1}$. Define the points $B_{1}$ and $C_{1}$ analogously. Suppose that the circumcentre of the triangle $A_{1} B_{1} C_{1}$ lies on the circumcircle of the triangle $A B C$. Prove that the triangle $A B C$ is right-angled. (Russia)

让与$A$相对的三角形$A B C$的外圆在点$A_{1}$处接触其边$B C$。类似地定义点 $B_{1}$ 和 $C_{1}$。假设三角形 $A_{1} B_{1} C_{1}$ 的外心位于三角形 $A B C$ 的外接圆上。证明三角形$A B C$ 是直角。 (俄罗斯)