Arithmetic of triangles

TL;DR

This paper explores the algebraic structure of a set of similar triangles with parallel sides, introducing novel operations and examining their geometric and arithmetic properties, including triangle dissection and vector sets.

Contribution

It introduces a new algebraic framework for describing similar triangles and their dissection, linking geometric configurations with algebraic operations in a novel way.

Findings

The set $\\mathbb{R}_2$ models similar triangles with parallel sides.

A new addition operation captures homothety and translation.

Dissection of triangles into 15 smaller triangles is described algebraically.

Abstract

In this paper, we consider a set of similar triangles with parallel sides, along with a set of points in the plane. It turns out that the set $\mathbb{R}_2= \{\pm <x >=\pm (x^2,x,1); x\in\mathbb{R} \}$ describes this set of triangles quite well. The set $\mathbb{R}_2$ is a subset of the ring $\mathbb{R}^3=\mathbb{R}\times\mathbb{R}\times\mathbb{R}= \{ (x,y,z) ; x,y,z\in\mathbb{R} \}$ with addition and multiplication defined coordinate-wise. The set $\mathbb{R}_2$ is equipped with two operations. Multiplication is inherited from the ring $\mathbb{R}^3$, while addition is a ternary operation that represents homothety and translation of elements in $\mathbb{R}_2$. However, the defined addition has its limitations. It turns out that, within this framework, the reduction of terms with different signs is not always possible. This leads to the distinction between an equation that is true in the arithmetic sense and one that is true in the geometric sense. A novel form of addition in $\mathbb{R}_2$ leads to intriguing properties of multiplication in $\mathbb{R}_2$, which are examined in a dedicated chapter. In the next section we use the construction of adding to describe the dissection of the triangle into 15 triangles of different sides. In the final two sections, we consider a set of two kinds of vectors, along with a set of points on the line. The set $\mathbb{R}_1= \{\pm <x >=\pm (x,1); x\in\mathbb{R} \}$ describes this set vectors quite well and it is a one-dimensional reduction of the set $\mathbb{R}_2$.