The Four Periodic Inscription Problem

We prove that four periodic curves in $\mathbb{C}$ inscribe a family of isosceles trapezoids for every angle $\theta \in (0,\pi )$, whose similarity types are determined by the signed areas between the quotients of the curves in the infinite cylinder. In the smooth case, the problem is identified with a Lagrangian intersection problem in $\mathbb{R}\times S^1\times \mathbb{C}$, while the continuous case follows by a convergence argument.