In an acute-angled triangle $ABC$, the circle with diameter $[AB]$ intersects the altitude drawn from vertex $C$ at a point $D$ and the circle with diameter $[AC]$ intersects the altitude drawn from vertex $B$ at a point $E$. Let the lines $BD$ and $CE$ intersect at $F$. Prove that $$AF\perp DE$$
Problem
Source: 2021 Turkey JBMO TST P1
Tags: geometry, circles, perpendicular lines, Turkey, Junior