Bisecting a triangle in a given direction

TL;DR

This paper investigates lines with a specified slope that bisect a triangle's area, providing a direct vector algebra method to determine such lines, building on previous geometric envelope characterizations.

Contribution

It introduces a direct vector algebra approach to find area-bisecting lines with a given slope in a triangle, simplifying prior geometric methods.

Findings

Derived a direct vector algebra formula for area-bisecting lines with a given slope.

Connected the set of bisecting lines to a 'deltoid' envelope shape.

Provided an efficient method for calculating bisecting lines in triangles.

Abstract

Given a triangle, what is the equation of the line which bisects its area and has a given slope? The set of all lines bisecting the area of a triangle has been elegantly determined as a certain 'deltoid' envelope and this gives an indirect method of solution. We find that vector algebra allows the equation to be written down rather directly and neatly.