5. (7 points) Two vertices, the incenter of the inscribed circle, and the intersection point of the altitudes of an acute triangle lie on the same circle. Find the angle at the third vertex.
Answer: $60^{\circ}$.

Solution. Consider triangle $ABC$, in which altitudes $AA_1$ and $BB_1$ are drawn. Let point $H$ be the orthocenter, and point $I$ be the incenter.