An Investigation of the Orthocenter of a Triangle, and the Nine
Point Circle
by Jamila K. Eagles
The Orthocenter of a circle is formed by the intersection of altitudes
from each vertex to its respective side. In Triangle ABC the orthocenter
is represented by the point H.
We can construct the nine point circle of triangle ABC by first findind
the nine points which are, the three feet of the altitudes, the three midpoints
of each side, and the three midpoints of each segment between the orthocenter
and each vertex. The nine points are selected in the picture below. The
center of the nine point circle N is located by finding the midpoint between
the orthocenter and the circumcenter of the triangle. This is also shown
below.
Now we can take a look at individual triangles formed by the orthocenter
of triangle ABC. First let's take a look at triangle ABH. We can construct
its nine point circle by selecting its nine points. In the figure below,
we see that the nine point circle formed is the same circle that was formed
by the nine points of triangle ABC.
Let's try to construct the nine point circle for triangle ACH. From
the figure below, once again we have the same circle.
Finally , constructing the nine point circle for triangle BCH, We see
that we get the same circle.
From my exploration I conclude that the nine point circle that is formed
by the original triangle ABC is the same nine point circle formed by the
individual triangles formed by the altitudes of the triangle that intersect
at the orthocenter. Although by demonstration, the statement seems true,
these pictures can not be considered a proof of this explanation.