2003 USAMO Problems/Problem 2
Problem
A convex polygon 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
are rational numbers. Prove that the lengths of all sides of all polygons in the dissection are also rational numbers.
Solution
When 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
within the polygon be
and
. Since
is a convex quadrilateral with sides and diagonals of rational length, we consider it in isolation.
By the Law of Cosines, , which is rational. Similarly,
is rational, as well as
. It follows that
is rational. Since
is rational, this means that
is rational. This implies that
is rational. This means that for some rational number
,
, which is, of course, rational. It follows that
and
both have rational length, as desired.
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.