2021 USAMO 第 6 题

Let $ABCDEF$ be a convex hexagon satisfying $\overline{AB} \parallel \overline{DE}$, $\overline{BC} \parallel \overline{EF}$, $\overline{CD} \parallel \overline{FA}$, and

$$AB \cdot DE = BC \cdot EF = CD \cdot FA.$$

Let $X$, $Y$, and $Z$ be the midpoints of $\overline{AD}$, $\overline{BE}$, and $\overline{CF}$. Prove that the circumcenter of $\triangle ACE$, the circumcenter of $\triangle BDF$, and the orthocenter of $\triangle XYZ$ are collinear.

设 $ABCDEF$ 为凸六边形，满足 $\overline{AB} \parallel \overline{DE}$、$\overline{BC} \parallel \overline{EF}$、$\overline{CD} \parallel \overline{FA}$，并且

$$AB \cdot DE = BC \cdot EF = CD \cdot FA。$$

设$X$、$Y$和$Z$为$\overline{AD}$、$\overline{BE}$和$\overline{CF}$的中点。证明$\triangle ACE$的外心、$\triangle BDF$的外心和$\triangle XYZ$的垂心共线。