2003 USAMO 第 2 题

A convex polygon $\mathcal{P}$ in the plane is dissected into smaller convex polygons by drawing all of its diagonals. The lengths of all sides and all diagonals of the polygon $\mathcal{P}$ are rational numbers. Prove that the lengths of all sides of all polygons in the dissection are also rational numbers.

平面上的凸多边形 $\mathcal{P}$ 通过绘制其所有对角线将其分割成更小的凸多边形。多边形 $\mathcal{P}$ 的所有边长和所有对角线的长度都是有理数。证明剖分中所有多边形的所有边长也是有理数。