The solution will consist of two parts. In the first part, we will prove that the points $C$, $C_1$, $C_2$ are collinear using Torrichelli point. In the second part we will be proving that $CC_1=CC_2$ using rotations Part 1: Proving that $C$, $C_1$ and $C_2$ are collinear Let $AA_1\cap BB_1=T$. Then $T$ is the Torrichelli point of triangle $ABC$ and $T$ is the Torichelli point of triangle $A_1B_1C$. So $C$, $T$, $C_1$ are collinear. Analogously, points $C$, $C_2$ and $T$ are collinear. This means that points $C$, $C_1$, $C_2$ all lie on line $CT$, hence are collinear Part 2: Proving $CC_1=CC_2$ Let's prove that $AA_1=BB_1$. Consider rotation centered at point $C$ taking $B_1$ to $A$, then $B$ is taken to $AA_1$, meaning that the segment $BB_1$ is taken to $AA_1$. Analogously, rotating around $B$ you can get $CC_1=AA_1$. Thus, $AA_1=BB_1=CC_1$. Note that the same can be done with vertices of triangle $A_1B_1C_2$ proving $CC_2=AA_1=BB_1$. From this follows, that $CC_1=CC_2$

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