Problem

Source: Slovenia 1998 4th Grade P3

Tags: geometry, incenter, circumcircle, Triangle



In a right-angled triangle $ABC$ with the hypotenuse $BC$, $D$ is the foot of the altitude from $A$. The line through the incenters of the triangles $ABD$ and $ADC$ intersects the legs of $\triangle ABC$ at $E$ and $F$. Prove that $A$ is the circumcenter of triangle $DEF$.