Sharygin 2018, final round, grade 9, problem 4
Sharygin 2018, final round, grade 9, problem 4 is a plane geometry problem from the Sharygin Geometry Olympiad 2018, final round, grade 9, problem 4: circumcenters on a fixed circle.
Problem
Let $BC$ be a fixed chord of a given circle $\omega$. Let $A$ be a variable point on the major arc $BC$ of $\omega$. Let $H$ be the orthocenter of triangle $ABC$. Points $D$ and $E$ lying on lines $AB$ and $AC$ respectively are such that $H$ is the midpoint of segment $DE$. Let $O_A$ be the circumcenter of triangle $ADE$. Prove that all points $O_A$ lie on a fixed circle.
Source Sharygin Geometry Olympiad 2018, final round, grade 9, problem 4 · proposed by A. Mudgal (India)
The diagram
Drag any point and the whole construction follows; ▶ runs a point along its path and back. Tap a point to pin it, double-click to go back to the start. Effects redraws the figure in grayscale for print, in colour, or with the givens in focus.