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.



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