Problem

Source: All-Russian Olympiad 2012 Grade 10 Day 1, UZMO 2012

Tags: geometry, inradius, circumcircle, geometric transformation, reflection, perpendicular bisector, geometry unsolved



The inscribed circle $\omega$ of the non-isosceles acute-angled triangle $ABC$ touches the side $BC$ at the point $D$. Suppose that $I$ and $O$ are the centres of inscribed circle and circumcircle of triangle $ABC$ respectively. The circumcircle of triangle $ADI$ intersects $AO$ at the points $A$ and $E$. Prove that $AE$ is equal to the radius $r$ of $\omega$.