Napoleon's theorem for quadrilaterals
Here's an applet illustrating Quang Hung Tran's theorem from the article "A Napoleon-like Theorem for Quadrilaterals" (in the American Mathematical Monthly, December 2022, pp. 975–977). Given the purple quadrilateral, construct squares on opposite sides (red) and parallelograms (blue) on the other two sides. The sides of the parallelograms are perpendicular to and the same length as the opposite sides of the quadrilateral (so the parallelograms are congruent). The theorem then states that the centers of the two red squares and two blue parallelograms form a square (black).
The hypotheses of the theorem require that ABCD be convex and and . Are those hypotheses needed? Playing with the applet it seems like the black quadrilateral is always a square—even when the quadrilateral is nonconvex!