Eight-Point Circle Theorem

Let $ABCD$ be a Quadrilateral with Perpendicular Diagonals. The Midpoints of the sides ($a$, $b$, $c$, and $d$) 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 $a'$, $b'$, $c'$, and $d'$.

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.