In Katherine Knox's article "Billiard circuits in quadrilaterals" ([i]Amer. Math. Monthly[/i], 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).[br][br][b]Theorem:[/b] A quadrilateral has a billiard circuit if and only if it is cyclic and its interior contains the center of its circumcircle. [br][br][b]Theorem:[/b] 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.[br][br]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.