A little noticed right triangle

TL;DR

This paper explores the geometric properties of a specific triangle derived from a right triangle, deriving formulas and conditions under which the new triangle is also Pythagorean, with numerical examples included.

Contribution

It introduces a novel geometric construction involving circumcenters and the midpoint of the hypotenuse, analyzing when the resulting triangle is Pythagorean.

Findings

The triangle O1OO2 is similar to the original right triangle ABC.

Derived formulas for segment lengths in the constructed triangle.

Identified conditions for the constructed triangle to be Pythagorean.

Abstract

Given a right triangle ABC, with the ninety degree angle at A; consider the triangle O1OO2.Where the point O is the midpoint of the hypotenuseBC(and so the center of the triangle ABC's circumcircle), the point O1 being the triangle AOB's circumcenter, and the point O2 being the triangle AOC circumcenter. The triangle O1OO2 is actually a right triangle similar to ABC. In the early part of this work, some geometrical formulas are derived, including some lengths of segments from a nearby trapezoid. After that, the focus changes. We examine the case when ABC is a Pythagorean triangle, and find the precise conditions under which the triangle O1OO2 is also Pythagorean. A numerical table of values is also offered at the end of the paper.