Let $A B C$ be a scalene acute triangle. Let $B_{1}$ be the point on ray $\left[A C\right.$ such that $\left|A B_{1}\right|=\left|B B_{1}\right|$. Let $C_{1}$ be the point on ray $\left[A B\right.$ such that $\left|A C_{1}\right|=\left|C C_{1}\right|$. Let $B_{2}$ and $C_{2}$ be the points on line $B C$ such that $\left|A B_{2}\right|=\left|C B_{2}\right|$ and $\left|B C_{2}\right|=\left|A C_{2}\right|$. Prove that $B_{1}, C_{1}, B_{2}, C_{2}$ are concyclic.