Question 65
Let $ABC$ be a triangle and $D, E, F$ be the midpoints of arcs $BC, CA, AB$ on the circumcircle. Line $l_a$ passes through the feet of the perpendiculars from $A$ to $DB$ and $DC$. Line $m_a$ passes through the feet of the perpendiculars from $D$ to $AB$ and $AC$. Let $A_1$ denote the intersection of lines $l_a$ and $m_a$. Define points $B_1$ and $C_1$ similarly. Prove that triangles $DEF$ and $A_1B_1C_1$ are similar to each other.