Sharygin triangles and elliptic curves

TL;DR

This paper explores a special family of scalene triangles called Sharygin triangles, showing they are related to elliptic curves and proving the existence of infinitely many integer solutions.

Contribution

It introduces Sharygin triangles, characterizes them via elliptic curves, and proves the existence of infinitely many non-similar integer solutions.

Findings

Sharygin triangles are parametrized by an open subset of an elliptic curve.

There are infinitely many non-similar integer Sharygin triangles.

The family of these triangles exhibits rich algebraic structure.

Abstract

The paper is devoted to the description of family of scalene triangles for which the triangle formed by the intersection points of bisectors with opposite sides is isosceles. We call them Sharygin triangles. It turns out that they are parametrized by an open subset of an elliptic curve. Also we prove that there are infinitely many non-similar integer Sharygin triangles.