Imaginary triangles, Pythagorean theorems, and algebraic geometry

TL;DR

This paper introduces the concept of imaginary triangles with complex sides and angles, explores their parametrization via algebraic curves, and derives Pythagorean-like theorems using Cremona transformations.

Contribution

It extends classical triangle notions to complex geometry, providing a new algebraic framework and Pythagorean relations for imaginary triangles.

Findings

Parametrization of imaginary triangles by algebraic curves

Derivation of Pythagorean theorems for complex triangles

Application of Cremona transformations to find implicit equations

Abstract

We extend the notion of triangle to "imaginary triangles" with complex valued sides and angles, and parametrize families of such triangles by plane algebraic curves. We study in detail families of triangles with two commensurable angles, and apply the theory of plane Cremona transformations to find "Pythagorean theorems" for them, which are interpreted as the implicit equations of their parametrizing curves.