The common tangents to the circumcircle and an excircle of triangle $ABC$ meet $BC, CA,AB$ at points $A_1, B_1, C_1$ and $A_2, B_2, C_2$ respectively. The triangle $\Delta_1$ is formed by the lines $AA_1, BB_1$, and $CC_1$, the triangle $\Delta_2$ is formed by the lines $AA_2, BB_2,$ and $CC_2$. Prove that the circumradii of these triangles are equal.