We begin constructiing the nine point circle by finding the midpoints
of each side of the given triangle.
Now we find the next three points by constucting the altitudes of the
triangle. The feet of the altitude will be the next three points that we
are looking for. The feet of the altitudes are represented by the blue dots
on the triangle below. We know that the intersection of the altitudes will
form the orthocenter of this triangle.
Now that we have the orthocenter, our next three points should also be
easy to find. They are the midpoints of the segments formed from the othocenter
to each vertex of the triangle. These points are labeled X,Y, and Z. We
now have our nine points.
We now need to construct a circle that will include all nine points that
we have found. If we form a triangle with vertices X, Y, and Z. We know
that the point eqidistant from each vertex is the center of a circle that
passes through all three points, namely the circumcircle of triangle XYZ.
So we begin by constructing the circumcenter of this triangle.
Notice that if C is equidistant from three points in the nine point circle,
then C should also be equidistant from the other six points of the circle.
Drawing the circumcircle of triangle XYZ gives us our nine point cirlce
with center C.
The question becomes how does the circumcenter of this smaller triangle
relate to the circumcenter of the larger triangle. We begin by constructing
the circumcenter of the larger circle. It is represented in the figure below
by the point CL. The circumcenter of the smaller triangle (C), the orthocenter
of the larger triangle(H), and the circumcenter of the larger triangle(CL),
appear to form a straight line.
Next I constructed a segment from H to CL. It formed a segment connecting
H and CL and including C. Next I constucted thed midpoint of the segment.
For this particular triangle the midpoint was at C. So the center of our
nine point circle appears to be the midpoint between the Orthocenter and
the circumcenter of our larger triangle.
I also observed the position of C with different sizes and shapes of
our larger triangle. The center of the nine point circle appears to be the
midpoint of the segment between the orthocenter and the circumcenter of
a triangle. This was only a demonstration and can not be considered a proof.
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