MathDB

Let

{ABC}

be an acute angled triangle, let

{O}

be its circumcentre, and let

{D,E,F}

be points on the sides

{BC,CA,AB}

, respectively. The circle

{(c_1)}

of radius

{FA}

, centered at

{F}

, crosses the segment

{OA}

{A'}

and the circumcircle

{(c)}

of the triangle

{ABC}

again at

{K}

. Similarly, the circle

{(c_2)}

of radius

DB

, centered at

D

, crosses the segment

\left( OB \right)

{B}'

and the circle

{(c)}

again at

{L}

. Finally, the circle

{(c_3)}

of radius

EC

, centered at

E

, crosses the segment

\left( OC \right)

{C}'

and the circle

{(c)}

again at

{M}

. Prove that the quadrilaterals

BKF{A}',CLD{B}'

and

AME{C}'

are all cyclic, and their circumcircles share a common point.
Evangelos Psychas (Greece)

Problem Statement