Question 24
In scalene triangle $ABC$, let the feet of the perpendiculars from $A$ to $BC$, $B$ to $CA$, $C$ to $AB$ 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 $BC$, $CA$, $AB$. Show that the perpendiculars from $D$ to $AA_2$, $E$ to $BB_2$, and $F$ to $CC_2$ are concurrent.
(This problem was suggested by Ian Le.)
(This problem was suggested by Ian Le.)