14th Turkish Mathematical Olympiad

Let $\{M\}=DC\cap AK$ and $\{N\}=DC\cap BL$. Considering the powers of the point $M$ with respect to the circles $ADE$ and $AFC$ we obtain $$ME\cdot MD=MK\cdot MA=MF\cdot MC.$$ Since $DE=FC$, we conclude that $M$ is the midpoint of the line segment $CD$. Similarly, $$NE\cdot ND=NL\cdot NB=NF\cdot NC,$$ and $N$ is the midpoint of the line segment $CD$. Therefore, $M=N$. Now the equalities above give $$MK\cdot MA=ME\cdot MD=NE\cdot ND=NL\cdot NB=ML\cdot MB$$ which implies that $K,A,L$ and $B$ lie on a circle.