Simplices with equiareal faces

TL;DR

This paper investigates three-dimensional simplices with faces of equal area, proving that non-degenerate such simplices have congruent faces and classifying degenerate cases with equiareal faces using elementary geometry.

Contribution

It provides a simple geometric proof that non-degenerate simplices with equiareal faces have congruent faces and characterizes all degenerate cases.

Findings

Non-degenerate simplices with equiareal faces have congruent faces.

Degenerate simplices with equiareal faces are fully classified.

Elementary geometric proof simplifies understanding of these simplices.

Abstract

We study simplices with equiareal faces in the Euclidean 3-space by means of elementary geometry. We present an unexpectedly simple proof of the fact that, if such a simplex is non-degenerate, than every two of its faces are congruent. We show also that this statement is wrong for degenerate simplices and find all degenerate simplices with equiareal faces.