Theorem:In DABC, let
A', B' and C' be the midpoints of the opposite sides; let D, E and F be
the feet of the altitudes; let H be the orthocenter, and let a, b and
c be the midpoints of AH, BH and CH respectively. Then the nine
points A', B', C', a, b, c, D, E and F all lie on a circle.
This sketch demonstrates this theorem. The circle drawn is the
circle with center the midpoint of A'a and through A'.
Note that it goes through the other seven points, as expected.