The problem of finding the minimum weight triangulation of a planar point set is a classic problem in computational geometry. No known algorithm exists which can find this type of triangulation ``quickly" for any finite input set of points in the plane, and this problem was shown to be NP-hard by Mulzer and Rote in 2006. The addition of points to an input set can have surprising effects on minimum triangulation weight. We prove that interesting topology arises in this setting. This topology is due in part to the k-ellipse, a generalized ellipse with k distinct foci. We will also explore how these curves arise as part of the level-zero set of an interesting family of polynomials