Marden's Theorem and Symmetric Parametrizations of Triangles Circumscribed about Central Conics

Using generalized Chapple--Euler relation, we prove that a triangle of unit circumradius may be circumscribed about a central conic with foci \( a_1,a_2\in\mathbb C \) if and only if the major axis of the conic has length \( |1-\overline{a_1}a_2| \). This characterization applies uniformly to both types of central conics.Our main result shows that Marden's theorem yields a parametrization of every such Poncelet family. The elementary symmetric polynomials of the vertices of a circumscribed triangle admit explicit formulas in terms of the foci \( a_1,a_2 \) and a unimodular parameter \( \lambda \in \mathbb C \). As a consequence, the classical degree-3 Blaschke-product equation arises directly from Marden's theorem and remains valid without requiring the foci to lie inside the unit disk.The parametrization provides a unified framework for deriving geometric invariants of families of circumscribed triangles. Using Marden's theorem, we provide a short proof of a classical theorem which states that the circumcircle of a triangle formed by three tangents to a parabola passes through the focus. Finally, we establish a higher-degree symmetric parametrization for partial fractions with finite numbers of poles, leading to generalized Möbius-product equations. The associated residues satisfy a partition-of-unity identity extending the classical Marden weights, thereby placing Marden's theorem within a broader residue-theoretic framework and suggesting possible higher-degree analogues in Poncelet geometry.