2023 IMO 第 6 题

Let $A B C$ be an equilateral triangle. Let $A_{1}$ , $B_{1}$ , $C_{1}$ be interior points of $A B C$ such that $B A_{1} = A_{1}C$ , $C B_{1} = B_{1}A$ , $A C_{1} = C_{1}B$ , and

$$\angle B A_{1}C + \angle C B_{1}A + \angle A C_{1}B = 480^{\circ}.$$

Let $A_{2} = \overline{BC_{1}} \cap \overline{CB_{1}}$ , $B_{2} = \overline{CA_{1}} \cap \overline{AC_{1}}$ , $C_{2} = \overline{AB_{1}} \cap \overline{BA_{1}}$ . Prove that if triangle $A_{1}B_{1}C_{1}$ is scalene, then the circumcircles of triangles $AA_{1}A_{2}$ , $BB_{1}B_{2}$ , and $CC_{1}C_{2}$ all pass through two common points.

设$A B C$ 为等边三角形。令 $A_{1}$ 、 $B_{1}$ 、 $C_{1}$ 为 $A B C$ 的内点，使得 $B A_{1} = A_{1}C$ 、 $C B_{1} = B_{1}A$ 、 $A C_{1} = C_{1}B$ ，并且

$$\角度 B A_{1}C + \角度 C B_{1}A + \角度 A C_{1}B = 480^{\circ}。$$

设 $A_{2} = \overline{BC_{1}} \cap \overline{CB_{1}}$ 、 $B_{2} = \overline{CA_{1}} \cap \overline{AC_{1}}$ 、 $C_{2} = \overline{AB_{1}} \cap \overline{BA_{1}}$ 。证明如果三角形 $A_{1}B_{1}C_{1}$ 是不等边三角形，则三角形 $AA_{1}A_{2}$ 、 $BB_{1}B_{2}$ 和 $CC_{1}C_{2}$ 的外接圆都经过两个公共点。