Olympiad problems

1998 Belarus Team Selection Test, 4

The altitudes through the vertices $ A,B,C$ of an acute-angled triangle $ ABC$ meet the opposite sides at $ D,E, F,$ respectively. The line through $ D$ parallel to $ EF$ meets the lines $ AC$ and $ AB$ at $ Q$ and $ R,$ respectively. The line $ EF$ meets $ BC$ at $ P.$ Prove that the circumcircle of the triangle $ PQR$ passes through the midpoint of $ BC.$

2023 EGMO, 2

Tags: easy geometry , geometry

We are given an acute triangle $ABC$. Let $D$ be the point on its circumcircle such that $AD$ is a diameter. Suppose that points $K$ and $L$ lie on segments $AB$ and $AC$, respectively, and that $DK$ and $DL$ are tangent to circle $AKL$. Show that line $KL$ passes through the orthocenter of triangle $ABC$.

2019 Romania Team Selection Test, 2

Tags: trigonometry , geometry , circumcircle , triangle , easy geometry

The altitudes through the vertices $ A,B,C$ of an acute-angled triangle $ ABC$ meet the opposite sides at $ D,E, F,$ respectively. The line through $ D$ parallel to $ EF$ meets the lines $ AC$ and $ AB$ at $ Q$ and $ R,$ respectively. The line $ EF$ meets $ BC$ at $ P.$ Prove that the circumcircle of the triangle $ PQR$ passes through the midpoint of $ BC.$

1997 IMO Shortlist, 18

Tags: geometry , trigonometry , triangle , easy geometry , circumcircle

The altitudes through the vertices $ A,B,C$ of an acute-angled triangle $ ABC$ meet the opposite sides at $ D,E, F,$ respectively. The line through $ D$ parallel to $ EF$ meets the lines $ AC$ and $ AB$ at $ Q$ and $ R,$ respectively. The line $ EF$ meets $ BC$ at $ P.$ Prove that the circumcircle of the triangle $ PQR$ passes through the midpoint of $ BC.$