  The subject of this article is the proof that the Heron triangle with three
integer medians does not exist. The article provides proofs of three lemmas. As
a result, the infinite descent method proved that the Heron triangle with three
integer medians does not exist. In the process of proving Lemma 3, the fact was
used that there is no right-angled Heron triangle with three integer medians.
And then it is proved that there is no Heron triangle with three natural
medians. The relevance of this article lies in the fact that the problem under
study is one of the unsolved problems of number theory.
