Isosceles Trapezoid Peg Problem

Let a Jordan curve be a simple closed curve $\gamma\subset\mathbb{R}^2$. An isosceles trapezoid is a quadrilateral whose vertices are the intersection set of two parallel lines and a circle; a curve inscribes a polygon $P$ if it contains four points forming a polygon similar to $P$ under an orientation-preserving similarity. Isosceles Trapezoid Peg Problem. Every Jordan curve inscribes every isosceles trapezoid. This is a generalisation of the Square Peg Problem and remains an open question in the stated generality, although the paper establishes the claim for smooth Jordan curves and additional nonsmooth cases.