The Anticenter and Reflection Line of a Cyclic Quadrilateral

The Simson point S of a quadrilateral Q is the point for which the pedal polygon of Q with respect to S degenerates into a single line, called the Simson line. If we reflect the Simson point in the lines containing the sides of Q, then we get another line that is parallel to the Simson line. We refer to this second line as the Reflection line of S. Ferrarello, Mammana, and Pennisi have conjectured that if Q is a cyclic quadrilateral that does not have parallel sides, then the reflection line of S passes through the anticenter of Q. We give a positive answer to this conjecture. We also give characterizations using the reflection line for a convex quadrilateral to be cyclic or to be semi-symmetric.