Is every triangle a trajectory of an elliptical billiard?

The paper proves that for every triangle there exists a unique ellipse for which the triangle is a 3-periodic billiard trajectory, and it gives two proofs: (i) via Ceva + Marden (Theorem 5.1) and (ii) via Pascal + Menelaus (a synthetic construction). Both are explicitly presented and justified in the uploaded manuscript, including the remark that the seemingly overdetermined six conditions (three points and three tangents) nonetheless yield a unique ellipse. The candidate solution follows the same two-pronged strategy and is correct in structure and conclusion. The only substantive flaw is a wrong explicit formula for the Ceva weights (it uses linear side combinations instead of the squared-side expression derived in Lemma 2.6), but that inaccuracy does not affect the existence/uniqueness argument. Key steps in the paper include: the excentral construction, the Ceva ratios for the orthic triangle, the Siebeck–Marden conic tangency theorem, and the Pascal/Menelaus collinearity result; all are consistent with the model’s outline. See Theorem 5.1 and its two proofs, relying on Ceva+Marden and on Pascal+Menelaus, respectively ; the discussion that six conditions still admit a unique ellipse ; the Ceva ratios and explicit formula (2.2) for the feet of altitudes ; the Siebeck–Marden theorem statement used in the construction ; and the Pascal corollaries used to fix the third tangent line .