Let be a Quadrilateral with Perpendicular Diagonals. The Midpoints of the sides (, , , and ) determine a Parallelogram (the Varignon Parallelogram) with sides Parallel to the Diagonals. The eight-point circle passes through the four Midpoints and the four feet of the Perpendiculars from the opposite sides , , , and .
See also Feuerbach's Theorem
References
Brand, L. ``The Eight-Point Circle and the Nine-Point Circle.'' Amer. Math. Monthly 51, 84-85, 1944.
Honsberger, R. Mathematical Gems II.
Washington, DC: Math. Assoc. Amer., pp. 11-13, 1976.