On Some Rational Triangles

Abstract

We prove that every set of n ≥ 3 points in \({\mathbb{R}^2}\) can be slightly perturbed to a set of n points in \({\mathbb{Q}^2}\) so that at least 3(n − 2) of mutual distances between those new points are rational numbers. Some special rational triangles that are arbitrarily close to a given triangle are also considered. Given a triangle ABC, we show that for each e > 0 there is a triangle A′B′C′ with rational sides and at least one rational median such that |AA′|, |BB′|, |CC′| < e and a Heronian triangle A′′B′′C′′ with three rational internal angle bisectors such that \({A^{\prime\prime}, B^{\prime\prime}, C^{\prime\prime} \in \mathbb{Q}^2}\) and |AA′′|, |BB′′|, |CC′′| < e.