Integer-Sided Triangles with integral medians

TL;DR

This paper proves that no triangle with integer sides can have all medians also integer, provides conditions for certain triangles to have two integral medians, and describes parametric families of special integer-sided triangles.

Contribution

It establishes the impossibility of all medians being integral in integer-sided triangles and characterizes specific families with two integral medians.

Findings

No integer-sided triangle has all three medians integral.

Conditions for nonisosceles triangles to have two integral medians.

Parametric descriptions of special families of integer-sided triangles.

Abstract

In this work, we prove that any triangle whose three sidelengths are integers, cannot have all of its three medians also having integral lengths.This is done in Proposition 2.In Section 5, we give precise(i.e.necessary and sufficient)conditions for a nonisosceles, integer-sided triangle to have two integral medians.In Section 3, we offer parametric descriptions of three special families of integer-sided triangles.The first family consists of all Pythagorean triangles whose medians to their hypotenuses is integral. The other two medians (to the two legs)of any Pythagorean triangle are irrational, a proof of this fact can be found in reference 4 of this paper.The second family consists of all integer-sided isosceles triangles, whose only integral median is the one contained between the two sides of equal length. Finally, the third family consists of all isosceles integer-sided triangles with the two medians of equal length;having integer length.