Equidissections of darts

TL;DR

This paper explores the geometric properties of a specific nonconvex quadrilateral called a dart, focusing on its ability to be dissected into equal-area triangles, especially when the number of triangles is odd.

Contribution

It introduces the concept of the dart $D(a)$ and investigates conditions under which it can be dissected into an odd number of equal-area triangles, extending previous work on even dissections.

Findings

Darts can be dissected into any even number of equal-area triangles.

Conditions for dissecting darts into an odd number of equal-area triangles are identified.

The study expands understanding of geometric dissections of nonconvex quadrilaterals.

Abstract

We define the dart $D(a)$ to be the nonconvex quadrilateral whose vertices are $(0,1), (1,1), (1,0), (a,a)$ (in counterclockwise order), with $a>1$. Such a dart can be dissected into any even number of equal-area triangles. Here we investigate darts that can be dissected into an odd number of equal-area triangle.