On Isosceles Triangles and Related Problems in a Convex Polygon

TL;DR

This paper investigates the maximum number of specific isosceles triangles and regular polygons that can be formed within a convex n-gon, providing bounds, conjectures, and proofs for these geometric configurations.

Contribution

It establishes new bounds on the number of isosceles triangles and regular polygons in convex n-gons, proves a conjecture for certain cases, and offers a concise proof for vertex distance sums.

Findings

Maximum isosceles triangles with two sides of unit length is approximately n^2/2

Bound on the number of regular k-gons is tight and proportional to n/k

Vertex distance sum bounds depend on the perimeter and number of vertices

Abstract

Given any convex $n$-gon, in this article, we: (i) prove that its vertices can form at most $n^2/2 + \Theta(n\log n)$ isosceles trianges with two sides of unit length and show that this bound is optimal in the first order, (ii) conjecture that its vertices can form at most $3n^2/4 + o(n^2)$ isosceles triangles and prove this conjecture for a special group of convex $n$-gons, (iii) prove that its vertices can form at most $\lfloor n/k \rfloor$ regular $k$-gons for any integer $k\ge 4$ and that this bound is optimal, and (iv) provide a short proof that the sum of all the distances between its vertices is at least $(n-1)/2$ and at most $\lfloor n/2 \rfloor \lceil n/2 \rceil(1/2)$ as long as the convex $n$-gon has unit perimeter.