IMO 2019 Problem 6
IMO 2019 Problem 6 is a plane geometry problem from the International Mathematical Olympiad 2019, Problem 6: the incircle and the line through D perpendicular to EF.
Problem
Let $I$ be the incentre of acute triangle $ABC$ with $AB \neq AC$. The incircle $\omega$ of $ABC$ is tangent to sides $BC$, $CA$, and $AB$ at $D$, $E$, and $F$, respectively. The line through $D$ perpendicular to $EF$ meets $\omega$ again at $R$. Line $AR$ meets $\omega$ again at $P$. The circumcircles of triangles $PCE$ and $PBF$ meet again at $Q$. Prove that lines $DI$ and $PQ$ meet on the line through $A$ perpendicular to $AI$.
Source International Mathematical Olympiad 2019, Problem 6
The diagram
Drag any point and the whole construction follows. 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.