ASSIGNMENT #4 - The Nine-Point Circle
Author : Elizabeth Gore

This particular investigation will explore the existence of the Nine-Point Circle.  To adequately complete this task we must first take a quick look at the definitions and circumcircles of each of these for the same original triangle:  Medial Traingle, Orthic Triangle, and the Orthocenter Mid-segment Triangle.  After overlaying the circumcircles of each of the triangles listed above, you will result with  one beautifully presented circle, the Nine-Point Circle.  Each of the triangle's three vertices makes up one of the nine points on the circle. Finally at the end of the investigation I will provide a proof for the existance of the Nine-Point Circle, so that you do not have to only take my word on it.

The Medial Triangle

The Medial Triangle DEF of a given triangle ABC is constructed by connecting the midpoints of the three sides of ABC. The circumcircle is constructed by first finding the intersection of two perpendicular bisectors, the circumcenter, which is the center of you circle. Then you will use the distance from the circumcenter to a vertex of the triangle as your radius. This process of finding the circumcirle is the same for each of the three triangles we will discuss.


In this graph the centroid, intersection of the the medians of triangle ABC, is labled G. The endpoints of these medians are the midpoints of the sides AB, BC, and CA. Connecting these endpoints forms the medial traingle, which we have labled DEF.

The circumcenter of triangle DEF is labled as C1, since it is the first circumcenter we have found.

The Orthic Triangle

The orthic triangle of ABC is formed by finding the three altitudes. The altitude extends from each vertex of the traingle ABC to the opposite side at a right angle. The point on each side of the triangle where these lines intersect form the vertices of the orthic triangle pictured below.

The new found orthic triangle is labled JKL. Within this triangle we have found the orthocenter, labled H, which is the intersection of these earlier stated altitudes.

Once again we use our former method of finding the circumcircle on our new triangle JKL.

The Orthocenter Mid-Segment Triangle

The orthocenter mid-segment triangle is a bit more difficult to understand, because it involves finding both the orthocenter and then the midpoints between the orthocenter and the original vertices. So, to begin with you start your process as you would with the orthic triangle, by finding the orthocenter, H, (intersections of altitudes). Once you have done so, mark the midpoints between the orthocenter and each vertex, or HA, HB, HC. These midpoints have been labled R, S, T, and will be the vertices of your new triangle.

After connecting the midpoints, we have created the new triangle RST, which is the orthocenter mid-segment triangle. The circumcircle can then be constructed, just as before.

Overlaying the Circles.

If you look closely at the above triangles ABC, you will notice that they are the same triangle. We have found three new triangles with which the circumcircle of each lies on the vertices of the respective triangles. Let's look and see what happens when we put all of the NEW triangles together within ABC.

What has happened? Well, to begin with each of our circumcenters (C1, C2, C3) have overlain each other to become one center, C. Since our circumcenters are the same, we sould be able to deduce that the circumcircles will be the same as well. Here is a picture.

So, now we have

The Nine-Point Circle!!

Here is a cleaner picture without the triangle construction lines.

Think that thsi could just be coincidence? Check out the PROOF (adapted from a proof on the website of the Computer Science and Mathematics Depatment of Arkansas State University)

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