The eight point circle was first discovered by Louis Brand of
Cincinnati in
1944. This theorem can be useful in proving the far more famous
nine point circle theorem.
D


P  O
I 
 H
A_________________________C
X 
 G
F 

M  N




B
Above illustrates quadrilateral ABCD, its perpendicular diagonals, the eight points, and centroid X of ABCD.
The theorem
For a
quadrilateral whose
diagonals are
perpendicular, the
midpoints M, N, O, P of each sides, and the feet F, G, H, I of perpendiculars from the midpoints to the opposite sides all lie on a
circle centered at the
centroid X of the quadrilateral.
To further explain, midpoint M, the
midpoint of A and B, is defined as (1/2) (A + B) where "+" is the addition of
vectors.
Foot F is the
perpendicular intersection point of line
l_{AB} that passes through A and B, and line
l_{O} that passes through point O.
The
centroid X is defined as (1/4)(A + B + C + D).
The proof
Quadrilateral MNOP is what is known as a Varignon
parallelogram. The Varignon parallelogram of a quadrilateral is the parallelogram that forms from the midpoints of the sides of the original quadrilateral. The midpoints form a parallelogram because the new sides are parallel to the original quadrilateral's diagonals.
Since it is given that the diagonals are
perpendicular, the
Varignon parallelogram is rectangular.
The
theorem of Thales (pronounced: "tallies") states that
for a circle which has PN as a diagonal, O lies on the circle
iff angle PON is perpendicular. Since MNOP is a rectangle, both O and M lie on such circle. Since PN is the diagonal of this circle, the center is the midpoint of PN, or the midpoint of (1/2)(A + D) and (1/2)(B + C), which is (1/4)(A + B + C + D), the
centroid of quadrilateral ABCD.
Since the diagonals of MNOP are equal in length and bisect each other, MO is also a diagonal of the circle that contains points M, N, O, P.
∠ PGN is perpendicular by construction, hence point G also lies on the circle by the
theorem of Thales.
By reasons of symmetry it follows that all points M, N, O, P, F, G, H, I lie on the same circle.
QED
Source:
http://cuttheknot.com/Curriculum/Geometry/EightPointCircle.html