2012 IMO Shortlist G5

Let $A B C$ be a triangle with $\angle B C A=90^{\circ}$, and let $C_{0}$ be the foot of the altitude from $C$. Choose a point $X$ in the interior of the segment $C C_{0}$, and let $K, L$ be the points on the segments $A X, B X$ for which $B K=B C$ and $A L=A C$ respectively. Denote by $M$ the intersection of $A L$ and $B K$. Show that $M K=M L$.

令$A B C$ 为$\angle B C A=90^{\circ}$ 的三角形，并令$C_{0}$ 为从$C$ 开始的高度的脚。选取线段$C C_{0}$内部的点$X$，令$K、L$分别为线段$A X、B X$上的点，其中$B K=B C$和$A L=A C$。用 $M$ 表示 $A L$ 和 $B K$ 的交集。证明 $M K=M L$。