Problem: In scalene triangle $ABC$, let the feet of the perpendiculars
from $A$ to $\ol{BC}$, $B$ to $\ol{CA}$, $C$ to $\ol{AB}$
be $A_1, B_1, C_1$, respectively.
Denote by $A_2$ the intersection of lines $BC$ and $B_1C_1$.
Define $B_2$ and $C_2$ analogously.
Let $D$, $E$, $F$ be the respective midpoints
of sides $\ol{BC}$, $\ol{CA}$, $\ol{AB}$.
Show that the perpendiculars from $D$ to $\ol{AA_2}$,
$E$ to $\ol{BB_2}$ and $F$ to $\ol{CC_2}$ are concurrent.

Solution: We claim that they pass through the orthocenter $H$.
Indeed, consider the circle with diameter $\ol{BC}$,
which circumscribes quadrilateral $BCB_1C_1$ and has center $D$.
Then by Brocard theorem, $\ol{AA_2}$ is the polar of line $H$.
Thus $\ol{DH} \perp \ol{AA_2}$.