5 problems

Ellipse conjecture. The set of all interior points $D$ of the triangle $ABC$ satisfying this condition is an ellipse.

Let a pair of ellipses admit Poncelet 3-periodics, and consider the loci of the excenters of the triangles in the family. The pair is poristic when the associated Poncelet family h…

Let a pair of ellipses admit Poncelet 3-periodics, and let $X_1$ denote the incenter of a member of the family. The pair is confocal when its two ellipses share the same foci. Ince…

Let a pair of ellipses admit Poncelet 3-periodics, and let $X_1$ denote the incenter of a member of this family. If the pair is concentric at $O$, call $O$ the common center; other…

Let a triangle center be defined by barycentric coordinates that are rational functions of the squares of the triangle's sidelengths, and consider its locus over the relevant famil…