A cornucopia of pythagorean triangles

TL;DR

This paper investigates conditions under which specific right triangles formed by two externally tangent circles with integer radii are Pythagorean, providing precise formulas for the radii that produce such triangles.

Contribution

It derives exact conditions on the integer radii of tangent circles to generate Pythagorean right triangles from their geometric configuration.

Findings

Identifies necessary conditions for R1 and R2 for Pythagorean triangles

Provides explicit formulas for radii R1 and R2

Characterizes all such Pythagorean configurations

Abstract

Consider two circles, externally tangential,and with integer radii R1, R2; and with R1>R2.The two circles have three tangent lines in common, one of them being T1T2. If M is the midpoint of T1T2, and K the point of intersection of the lines C1C2 and T1T2;then 16 right triangles are formed(C1 and C2 are the two circle centers), see Figure 1.In Section 6 of this paper, we find the precice form the two integers R1 and R2 must have, in order that the sixteen aforementioned right triangles be Pythagorean.