Billiard circuits in quadrilaterals
In Katherine Knox's article "Billiard circuits in quadrilaterals" (Amer. Math. Monthly, October 2023), she proved the following theorems about billiard circuits (a periodic billiard trajectory that bounces off each side consecutively one time before closing up).
Theorem: A quadrilateral has a billiard circuit if and only if it is cyclic and its interior contains the center of its circumcircle.
Theorem: In a cyclic quadrilateral which contains a billiard circuit, there exists infinitely many billiard circuits, one for each point in the interior of any side for which neither of the opposite two angles are obtuse.
These theorems are illustrated in the following applet. You are able to move the points A, B, C, and D around the circle to get various cyclic quadrilaterals. If the points are arranged so that angles D and C are acute, then the trajectory leaving from F (a point that is also movable) is cyclic.