The eight point circle was first discovered by Louis Brand of Cincinnati
. This theorem can be useful in proving the far more famous nine point circle theorem
P | O
M | N
Above illustrates quadrilateral ABCD, its perpendicular diagonals, the eight points, and centroid X of ABCD.
For a quadrilateral
, 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 vector
Foot F is the perpendicular
intersection point of line
that passes through A and B, and line lO
that passes through point O.
X is defined as (1/4)(A + B + C + D).
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
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.