  Richard Guy asked the following question: can we find a triangle with
rational sides, medians, and area? Such a triangle is called a \emph{perfect
triangle} and no example has been found to date. It is widely believed that
such a triangle does not exist. Here we use the setup of Solymosi and de Zeeuw
about rational distance sets contained in an algebraic curve, to show that for
any angle $0<\theta < \pi$, the number of perfect triangles with an angle
$\theta$ is finite. A \emph{rational median set} $S$ is a set of points in the
plane such that for every three non collinear points $p_1,p_2,p_3$ in $S$ all
medians of the triangle with vertices at $p_i$'s have rational length. The
second result of this paper is that no irreducible algebraic curve defined over
$\mathbb{R}$ contains an infinite rational median set.

Combinatorics