Problem

Source: China Southeast Mathematical Olympiad

Tags: geometry



Let $ABC$ be an acute-angled triangle. In $ABC$, $AB \neq AB$, $K$ is the midpoint of the the median $AD$, $DE \perp AB$ at $E$, $DF \perp AC$ at $F$. The lines $KE$, $KF$ intersect the line $BC$ at $M$, $N$, respectively. The circumcenters of $\triangle DEM$, $\triangle DFN$ are $O_1, O_2$, respectively. Prove that $O_1 O_2 \parallel BC$.