Problem

Source: 2015 Taiwan TST Round 1 Quiz 3 Problem 2

Tags: Taiwan, geometry, Taiwan TST 2015



Given a triangle $ABC$, $A',B',C'$ are the midpoints of $\overline{BC},\overline{AC},\overline{AB}$, respectively. $B^*,C^*$ lie in $\overline{AC},\overline{AB}$, respectively, such that $\overline{BB^*},\overline{CC^*}$ are the altitudes of the triangle $ABC$. Let $B^{\#},C^{\#}$ be the midpoints of $\overline{BB^*},\overline{CC^*}$, respectively. $\overline{B'B^{\#}}$ and $\overline{C'C^{\#}}$ meet at $K$, and $\overline{AK}$ and $\overline{BC}$ meet at $L$. Prove that $\angle{BAL}=\angle{CAA'}$