# How support lines touch an arc

## TL;DR

This paper proves a geometric property of simple polygonal arcs, showing they can have at most two pairs of support lines with a specific angular relationship, revealing a new structural constraint.

## Contribution

The paper introduces a novel theorem about the maximum number of support line pairs for polygonal arcs with a given angle difference.

## Key findings

- Each simple polygonal arc has at most two such support line pairs.
- Support line pairs are characterized by specific points on the arc.
- The result constrains the geometric configurations of polygonal arcs.

## Abstract

We prove that each simple polygonal arc {\gamma} attains at most two pairs of support lines of given angle difference such that each pair has s1 < s2 < s3 that {\gamma}(s1) and {\gamma}(s3) are on one such line and {\gamma}(s2) is on the other line.