Let $P$ be a point in the interior of the acute-angled triangle $ABC$. Prove that if the reflections of $P$ with respect to the sides of the triangle lie on the circumcircle of the triangle, then $P$ is the orthocenter of $ABC$.
Source: 2017 Romania JBMO TST1 p1
Tags: geometry, circumcircle, reflection, orthocenter
Let $P$ be a point in the interior of the acute-angled triangle $ABC$. Prove that if the reflections of $P$ with respect to the sides of the triangle lie on the circumcircle of the triangle, then $P$ is the orthocenter of $ABC$.