Nine-Point Circles, Cyclic Quadrilaterals, and the Anticenter
If four points form a cyclic quadrilateral, then the four nine-point circles concur at the anticenter of the cyclic quadrilateral. The four nine-point circles are all congruent with a radius of half that of the cyclic quadrilateral's circumcircle. The four nine-point centers are cyclic and lie on a circle congruent to the four nine-point circles that is centered at the anticenter of the cyclic quadrilateral.