A note on Erd\H{o}s's mysterious remark

TL;DR

This paper provides an alternative algebraic geometry-based proof that the only six-point set in the plane with all triangle subsets isosceles is a regular pentagon with its center, addressing Erdős's related questions.

Contribution

It introduces a novel algebraic geometry approach to a geometric problem, extending previous results and offering new insights into Erdős's questions.

Findings

The unique six-point configuration is the regular pentagon with its center.

All triangles formed by these six points are isosceles.

The proof method can be extended to related geometric configurations.

Abstract

We give an alternative proof of the statement, by using elimination from algebraic geometry, that the only set $S\subset\mathbb{R}^2$, $\left|S\right|=6$ such that all subsets that form a triangle are isosceles triangles, is the regular pentagon with its center. Our proof can be extended to answer some related questions raised by Erd\H{o}s.