SHARYGIN 2021 Problem 20

The mapping $f$ assigns a circle to every triangle in the plane so that the following conditions hold. (We consider all nondegenerate triangles and circles of nonzero radius.)(a) Let $\sigma$ be any similarity in the plane and let $\sigma$ map triangle $\Delta_1$ onto triangle $\Delta_2$ . Then $\sigma$ also maps circle $f(\Delta_1)$ onto circle $f(\Delta_2)$ .(b) Let $A,B,C$ and $D$ be any four points in general position. Then circles $f(ABC),f(BCD),f(CDA)$ and $f(DAB)$ have a common point.

Prove that for any triangle $\Delta$ , the circle $f(\Delta)$ is the Euler circle of $\Delta$ .