Revisiting the quadrisection problem of Jacob Bernoulli

TL;DR

This paper thoroughly analyzes the quadrisection problem of triangles, providing a complete classification of such divisions and revealing unique properties of specific triangle types like isosceles triangles.

Contribution

It offers a comprehensive description of all quadrisections of a triangle, extending Euler's classical result and identifying unique cases such as the singular isosceles triangle with two quadrisections.

Findings

Complete classification of triangle quadrisections

Only one isosceles triangle has exactly two quadrisections

Extension of Euler's 1779 result on quadrisections

Abstract

Two perpendicular segments which divide a given triangle into 4 regions of equal area is called a quadrisection of the triangle. Leonhard Euler proved in 1779 that every scalene triangle has a quadrisection with its triangular part on the middle leg. We provide a complete description of the quadrisections of a triangle. For example, there is only one isosceles triangle which has exactly two quadrisections.