Publication Date

Spring 2026

Degree Type

Thesis

Degree Name

Master of Arts (MA)

Department

Mathematics and Statistics

Advisor

Jordan Schettler; Slobodan Simic; Yan Zhang

Abstract

We say a triangle △ is rational if its side lengths are rational. For a geometry of constant curvature κ, we say △ is a rational triangle if the generalized tangents of its side lengths are rational. We are able to parameterize rational triangles having the same inradius and semiperimeter on an plane curve, write a bijection between the rational points on this curve and triples of side lengths, and show that this curve is an elliptic curve. Then, we write a general transformation for our plane curve to short Weierstrass form to compute ranks and add points easily using Sage. We look at the rank of such curves and conjecture that the rank is always strictly positive, implying that there are infinitely many rational triangles sharing the same inradius and semiperimeter. We finish with examples of the point addition on a curve for κ = −1, 0, 1 and examples of the triangles generated by the points sharing semiperimeter and inradius.

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