MathDB

Given an acute triangle

{ABC}

, erect triangles

{ABD}

and

{ACE}

externally, so that

{\angle ADB= \angle AEC=90^o}

and

{\angle BAD= \angle 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

{K,L}

be the midpoints of

[ B{{C}_{1}} ]

and

{BC_1, CB_1}

, respectively. Prove that the circumcenters of the triangles

AKL,{{A}_{1}}{{B}_{1}}{{C}_{1}}

and

{DEA_1}

are collinear.
(Bulgaria)

Problem Statement