Let $A_0$ be the midpoint of side $BC$ of triangle $ABC$, and $A'$ be the point of tangency with this side of the inscribed circle. Let's construct a circle $ \omega$ with center at $A_0$ and passing through $A'$. On other sides we will construct similar circles. Prove that if $ \omega$ is tangent to the cirucmscribed circle on arc $BC$ not containing $A$, then another one of the constructed circles touches the circumcircle.