Question 167
An acute scalene triangle $ABC$ is inscribed in a circle $k$. The bisector of angle $\angle BAC$ meets side $BC$ at point $D$. Let $I$ be an arbitrary point on the segment $AD$, and let $H$ be the orthogonal projection of $I$ onto $BC$. Circle $\omega$ is centered at $I$ and passes through $H$, and $U$ is the internal homothety center of the circles $k$ and $\omega$. Prove that $U$ lies on a fixed line as $I$ moves on $AD$, and that $AI$ bisects the angle $HAU$.
