2003 USAMO Problems/Problem 2

Problem

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.

Solution

When $\mathcal{P}$ is a triangle, the problem is trivial. Otherwise, it is sufficient to prove that any two diagonals of the polygon cut each other into rational lengths. Let two diagonals which intersect at a point $\displaystyle P$ within the polygon be $\displaystyle AC$ and $\displaystyle BD$ . Since $\displaystyle ABCD$ is a convex quadrilateral with sides and diagonals of rational length, we consider it in isolation.

By the Law of Cosines, $\cos BAC = \frac{AB^2 + CA^2 - BC^2}{2AB \cdot CA}$ , which is rational. Similarly, $\displaystyle \cos CAD$ is rational, as well as $\displaystyle \cos BAD = \cos (BAC + CAD) = \cos BAC \cos CAD - \sin BAC \sin CAD$ . It follows that $\displaystyle \sin BAC \sin CAD$ is rational. Since $\displaystyle \sin^2 CAD = 1 - \cos^2 CAD$ is rational, this means that $\frac{\sin BAC \sin CAD}{\sin^2 CAD} = \frac{\sin BAC}{\sin CAD} = \frac{\sin BAP}{\sin PAD}$ is rational. This implies that $\frac{ AB \sin BAP}{ AD \sin PAD} = \frac{\frac{1}{2} AB \cdot AP \sin BAP}{\frac{1}{2} AP \cdot AD \sin PAD} = \frac{[BAP]}{[PAD]} = \frac{BP}{PD}$ is rational. This means that for some rational number $\displaystyle r$ , $\displaystyle (1+r)BP = BP + PD = BD$ , which is, of course, rational. It follows that $\displaystyle BP$ and $\displaystyle PD$ both have rational length, as desired.