Assignment #4
The Nine-Point Circle
by
Angela Wall
The Nine-Point circle for any triangle passes through the three midpoints of the sides, the three feet of the altitudes, and the three midpoints of the segments from the respective vertices to orthocenter. Construct the nine points, locate the center (N) and construct the nine point circle.
The Nine-Point circle is shown below. Constructing the circle by using the Geometer’s Sketchpad (GSP) can be relatively easy by using the definition of the circle. Finding the center of the circle, however, requires more exploration with the other centers of the triangle.
In the sketch above, the three midpoints of the sides of the triangle, the three feet of the altitudes, and the three midpoints of the segments from the respective vertices to orthocenter were found, nine points total. Selecting three consecutive points at a time, an arc through the three points was constructed. This process was continued until the arcs joined to make a circle, the Nine-Point circle.
There are also other ways to construct the Nine-Point circle, along with its center. To do this, three sub-triangles of the the original triangle need to be constructed. These sub-triangles are the medial triangle, the orthic triangle, and a dilated version of the original triangle.
For the dilated sub-triangle, the orthocenter of the original triangle needs to be found first. Marking the orthocenter as the center of dilation, dilate the original triangle by one-half. The vertices of the dilated triangle are the midpoints of the segments between the orthocenter and the vertices of the original triangle, three points that are required in the Nine-Point circle.
To construct the medial triangle, the midpoint for each side of the triangle needs to be found. Connecting the midpoints forms the medial triangle.
For the orthic triangle, the perpendiculars from each vertex need to be constructed. Defining a point at the base of each altitude on the triangle and connecting the points will result in the orthic triangle.
All of the sub-triangles have vertices that make up some part of the Nine-Point circle. If the circumcenter and circumcircle for each sub-triangle is found, one will notice that it is the same circle and point. This is the Nine-Point circle. Therefore the circumcenter of the sub-triangles, by definition of the circumcenter and circumcircle, will be the center of the Nine-Point circle.
To quickly find the Nine-Point circle, dilated the original triangle by one-half using the orthocenter of the triangle as the center of dilation. Find the the circumcenter and circumcircle of the dilated triangle. The circumcircle is the Nine-Point circle and the circumcenter is the center of the Nine-Point circle. Even more quickly, the circumcircle of the original triangle could be found and then dilated by a factor of one-half using the orthocenter as the center of dilation. The dilated circumcircle would be the Nine-Point circle.
