New Properties of Triangular Orbits in Elliptic Billiards

New invariants in the one-dimensional family of 3-periodic orbits in the elliptic billiard were introduced by the authors in “Can the Elliptic Billiard Still Surprise Us?” (2020), Math. Intelligencer, 42(1): 6–17, some of which were generalized to $N>3$. Invariants mentioned there included ratios of radii and/or areas, sum of angle cosines, and a special stationary circle. Here we present some of the proofs omitted there as well as a few new related facts.