12.1.1 Olympic geometry

Problem: On the sides of the triangle \( A B C \) points \( A_{1} \in \) \( [B C], A_{2} \in\left[A_{1} C\right], B_{1} \in[C A], B_{2} \in\left[B_{1} A\right] \) are marked. \( C_{1} \in[A B], C_{2} \in\left[C_{1} B\right] \), for which \[ \begin{array}{l} \frac{C A_{1}}{B C}=\frac{C B_{2}}{C A}=\frac{B C+C A}{A B+B C+C A}, \\ \frac{A B_{1}}{C A}=\frac{A C_{2}}{A B}=\frac{C A+A B}{A B+B C+C A}, \\ \frac{B C_{1}}{A B}=\frac{B A_{2}}{B C}=\frac{A B+B C}{A B+B C+C A} . \end{array} \] Prove that the intersection points of the lines \( A_{1} C_{2}, C_{1} B_{2} \) and \( B_{1} A_{2} \) line on the circumscribed circle of the triangle \( A B C \).