Periodic Inscription of Isosceles Trapezoids

We prove that a pair of continuous disjoint periodic curves in $\mathbb{C}$ inscribes an isosceles trapezoid with any similarity type. The case of smooth curves can be identified with a Lagrangian intersection problem for a pair of Lagrangian cylinders in $\mathbb{R}\times S^1\times\mathbb{C}$, and the continuous case follows from the smooth one by a standard convergence argument.