Cut the Cake (Challenge)

There are $N$ points in a plane (numbered $1$ through $N$). For each valid $i$, the coordinates of the $i$-th point are $(X_i, Y_i)$. No three of these points are collinear. Let's denote a line segment between points $u$ and $v$ by $(u,v)$. You are given a [Delaunay triangulation](https://en.wikipedia.org/wiki/Delaunay_triangulation) of those points, which consists of $M$ triangles (numbered $1$