# Loci of Points Inspired by Viviani's Theorem

## TL;DR

This paper explores geometric loci related to a triangle where the sum of distances or squared distances to sides remains constant, extending Viviani's theorem to new shapes like line segments and ellipses.

## Contribution

It introduces new geometric loci based on Viviani's theorem, including the characterization of these loci as line segments, the entire triangle, or ellipses.

## Key findings

- Loci of points with constant sum of distances form line segments or the entire triangle.
- Loci of points with constant sum of squared distances form ellipses.
- The work extends Viviani's theorem to broader geometric contexts.

## Abstract

We consider loci of points such that their sum of distances or sum of squared distances to each of the sides of a given triangle is constant. These loci are inspired by Viviani's theorem and its extension. The former locus is a line segment or the whole triangle and the latter locus is an ellipse.