Automedian triangle

In plane geometry, an automedian triangle is a triangle in which the lengths of the three medians (the line segments connecting each vertex to the midpoint of the opposite side) are proportional to the lengths of the three sides, in a different order. The three medians of an automedian triangle may be translated to form the sides of a second triangle that is similar to the first one.

The side lengths of an automedian triangle satisfy the formula 2a² = b² + c², analogous to the Pythagorean theorem characterizing right triangles as the triangles satisfying the formula a² = b² + c². That is, in order for the three numbers a, b, and c to be the sides of an automedian triangle, the sequence of three squared side lengths b², a², and c² should form an arithmetic progression.

If x, y, and z are the three sides of a right triangle, sorted in increasing order by size, and if 2x < z, then z, x + y, and y − x are the three sides of an automedian triangle. For instance, the right triangle with side lengths 5, 12, and 13 can be used to form in this way an automedian triangle with side lengths 13, 17, and 7.

The condition that 2x < z is necessary: if it were not met, then the three numbers a = z, b = x + y, and c = x − y would still satisfy the equation 2a² = b²+ c² characterizing automedian triangles, but they would not satisfy the triangle inequality and could not be used to form the sides of a triangle.

Consequently, using Euler's formula that generates primitive Pythagorean triangles it is possible to generate primitive integer automedian triangles as

with ${\displaystyle m}$ and ${\displaystyle n}$ coprime, ${\displaystyle m+n}$ odd and to satisfy the triangle inequality ${\displaystyle n<m<n{\sqrt {3}}}$  or  ${\displaystyle m>(2+{\sqrt {3}})n}$.

...
Wikipedia