Any cyclic quadrilateral can be inscribed in any closed convex smooth curve

TL;DR

This paper proves that any cyclic quadrilateral can be inscribed in any closed convex smooth curve, with the exception of rectangles which can be inscribed in any closed convex curve regardless of smoothness.

Contribution

The authors establish a general theorem for inscribing cyclic quadrilaterals in convex curves, extending previous results by removing smoothness constraints.

Findings

Any cyclic quadrilateral can be inscribed in any closed convex C^1-curve.

Rectangles can be inscribed in any closed convex curve without smoothness restrictions.

The smoothness condition is unnecessary for rectangles.

Abstract

We prove that any cyclic quadrilateral can be inscribed in any closed convex $C^1$-curve. The smoothness condition is not required if the quadrilateral is a rectangle.