Let $\omega$ denote the incircle of an acute, non-isosceles triangle $ABC$, which touches sides $BC$, $CA$, $AB$ at $A_1$, $B_1$, $C_1$ respectively. Let $I$ denote the center of $\omega$. On rays $IA_1$, $IB_1$, $IC_1$ take points $A_2$, $B_2$, $C_2$ such that $$IA_2 = AB_1 = AC_1,IB_2 = BA_1 = BC_1, IC_2 = CB_1 = CA_1$$Prove that there exist two distinct points lying on the circumcircle of triangles $AA_1A_2$, $BB_1B_2$ and $CC_1C_2$. (by JasonM#8428)