Question 156
Given an acute triangle $ABC$, erect triangles $ABD$ and $ACE$ externally, so that $\widehat{ADB} = \widehat{AEC} = 90^\circ$ and $\widehat{BAD} \equiv \widehat{CAE}$. Let $A_1 \in BC$, $B_1 \in AC$ and $C_1 \in AB$ be the feet of the altitudes of the triangle $ABC$, and let $K$ and $L$ be the midpoints of $[BC_1]$ and $[CB_1]$, respectively. Prove that the circumcenters of the triangles $AKL$, $A_1B_1C_1$ and $DEA_1$ are collinear.
