  Using Marden's Theorem from geometric theory of polynomials, we show that for
every triangle there is a unique ellipse such that the triangle is a billiard
trajectory within that ellipse. Since $3$-periodic trajectories of billiards
within ellipses are examples of the Poncelet polygons, our considerations
provide a new insight into the relationship between Marden's Theorem and the
Poncelet Porism, two gems of exceptional classical beauty. We also show that
every parallelogram is a billiard trajectory within a unique ellipse. We prove
a similar result for the self-intersecting polygonal lines consisting of two
pairs of congruent sides, named "Darboux butterflies". In each of three
considered cases, we effectively calculate the foci of the boundary ellipses.
