2005 IMO 第 1 题

Six points are chosen on the sides of an equilateral triangle $A B C$ .. $A_{1}$ $A_{2}$ on $B C$ $B_{1}$ $B_{2}$ on $C A$ and $C_{1}$ $C_{2}$ on $A B$ , such that they are the vertices of a convex hexagon $A_{1}A_{2}B_{1}B_{2}C_{1}C_{2}$ with equal side lengths. Prove that the lines $A_{1}B_{2}$ $B_{1}C_{2}$ and $C_{1}A_{2}$ are concurrent.

在等边三角形 $A B C$ .. $B C$ 上的 $A_{1}$ $A_{2}$ 的边上选择六个点，$C A$ 上的 $B_{1}$ $B_{2}$ 和 $A B$ 上的 $C_{1}$ $C_{2}$ ，这样它们就是凸六边形的顶点$A_{1}A_{2}B_{1}B_{2}C_{1}C_{2}$ 边长相等。证明行 $A_{1}B_{2}$ $B_{1}C_{2}$ 和 $C_{1}A_{2}$ 是并发的。