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.

中文翻译：

---

关于定角的完美三角形的数目

理查德·盖 (Richard Guy) 提出了以下问题：我们能否找到具有有理边、中位数和面积的三角形？这样的三角形被称为\emph{完美三角形}，迄今为止还没有发现任何例子。人们普遍认为这样的三角形不存在。这里我们使用 Solymosi 和 de Zeeuw 关于包含在代数曲线中的有理距离集的设置，来证明对于任何角度 $0<\theta <\pi$，具有角度 $\theta$ 的完美三角形的数量是有限的。\emph{有理中位数集} $S$ 是平面中的一组点，对于 $S$ 中的每三个非共线点 $p_1,p_2,p_3$，顶点位于 $p_i$' 的三角形的所有中位数s 有合理的长度。本文的第二个结果是定义在 $\mathbb{R}$ 上的不可约代数曲线不包含无限有理中值集。