Triangle in a brick

TL;DR

This paper investigates the maximum size of a regular triangle within a unit-volume brick with edge lengths exceeding 1/√2, identifying three optimal configurations with edge length √2.

Contribution

It establishes the largest possible regular triangles in a brick of volume 1 with specific edge length constraints and characterizes three optimal cases.

Findings

Identifies three optimal configurations with edge length √2.

Proves these are the largest regular triangles in the given brick.

Provides geometric characterization of the optimal cases.

Abstract

In this paper we shall investigate the following problem: Which is the largest regular triangle in a brick with volume 1 and having edge lengthes greater than $\frac{1}{\sqrt{2}}$? We prove that there are three optimal cases in all which the edge lengthes of the triangles are $\sqrt{2}$, respectively.