Equidissections of kite-shaped quadrilaterals

Abstract

Let $Q\left(a\right)$ be the convex kite-shaped quadrilateral with vertices $\left(0,0\right)$, $\left(1,0\right)$, $\left(0,1\right)$, and $\left(a,a\right)$, where $a>1∕2$. We wish to dissect $Q\left(a\right)$ into triangles of equal areas. What numbers of triangles are possible? Since $Q\left(a\right)$ is symmetric about the line $y=x$, $Q\left(a\right)$ admits such a dissection into any even number of triangles. In this article, we prove four results describing $Q\left(a\right)$ that can be dissected into certain odd numbers of triangles.