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EIGHT POINT CIRCLE

by Derelle McFarland

Another way to construct a circle is the eight point circle theorem. Given a quadrilateral ABCD with perpendicular diagonals we can construct eight points that form a circle. First construct the midpoints of side AB, BC, CD, and AD and the diagonals of the quadrilateral.

Next construct a parallelogram from the midpoints of the sides. This parallelogram is specifically known as the Varignon Parallelogram which I will explore later in this essay.

Now construct perpendiculars from each midpoint to the opposite side. Construct their points of intersection.

Now connect the eight points on the quadrilateral to form the eight point circle.

There are some very interesting properties of this circle. Let's first look at the area of quadrilateral ABCD and the area of the Varignon Parallelogram.

Notice the area of the Varignon Parallelogram(JIHK) is half the area of quadrilateral ABCD.

Another interesting characteristic deals with the perimeter of the
parallelogram and the sum of the diagonals of quadrilateral ABCD.

The perimeter of the parallelogram is equal to the sum of the diaonals of quadrilateral ABCD.

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