There is no Heron triangle with three rational medians

TL;DR

This paper proves that no Heronian triangle can have three integer medians, establishing a universal identity and showing such triangles can only have six integer elements among sides, medians, and area.

Contribution

It introduces a new lemma and a universal identity that together prove the nonexistence of Heronian triangles with three integer medians.

Findings

Heronian triangles with three integer medians do not exist.

Among the seven elements of a triangle, only six can be integers.

A universal identity valid for all triangles is established.

Abstract

The relevance of this paper lies in the fact that it resolves two previously unsolved open problems. In the first part of the paper, a new lemma is proved, from which it follows that if there exists a triangle with integer sides and medians, then there necessarily exists another triangle, not similar to it, with the same properties. In other words, such triangles can exist only in pairs. In the second part of the paper, by transforming known formulas, a new theorem is established in the form of a universal identity valid for all triangles. The focus of this theorem is the proof of the nonexistence of Heronian triangles with three integer medians. We arrive at the conclusion that, among the seven elements of a triangle (three sides, three medians, and the area), only six can be integers. It should be noted that if the above universal identity is considered as a Diophantine equation, then such a Diophantine equation does have solutions. As examples, one may consider the following pair of triangles. A triangle with sides 146,102,52, area 1680, and two integer medians 35 and 97. Also, according to the conditions of the above lemma, there exists another triangle with medians 219,153,78, area 5040, and two integer sides 70 and 194. This version of the article is different from the previous ones because everything has been simplified to the level of a straight-A high school student.