Assignment 8: The Nine Point Circle concerning Altitudes and Otrhocenters

By: David Drew

EMAT 6680, J. Wilson

We want to investigate the nine point circle, its history, and eventually make a conjecture and give a proof of our assumption. So let’s begin with a triangle and show how we get the nine point circle. Given any triangle ABC we start by finding all the altitudes of the triangle. Theses lines are the lines that pass through a vertex and are perpendicular to the side opposite of the given vertex. We label their intersection H, or the orthocenter. We now make segments AH, BH, and CH and find the midpoints of all those segments. Next we find the midpoints of AB, BC, and AC and construct a triangle out of those three points. We make a circumcircle from our ‘midpoints’ triangle and we have the Nine Point circle. The points on the circle are as follows: the midpoints of the segments of our original triangle, the intersection of our altitudes and the sides of our triangle, and the midpoints of AH, BH, and CH. Now that we know how to build the Nine Point Circle can we make any conjectures and back those up with proofs?

Click on Nine Point to move or shift the circle around.

We’ll go with an interesting observation from my own investigation and try to realize how and why it is true. The four circumcircles of points in an orthocentric system taken three at a time have equal radius.

Now that we’ve stated this theory we need to go back and define a few vocabulary words. An orthocentric system is given by the picture below and here on GSP. It’s simply the four sets of triangles given by one triangle A, B, C and its orthocenter, H. According to this picture there are four such different combinations, and all of their circumcircles have an equal radius. To condense the picture I’ve located all the circumcircles onto the original picture to give us this image. Now comes the trick of showing that all these circles have the same radius. Click on Circumcircles to look at GSP and the measurements. But since we can’t use GSP as a tool of proof then we need to show some geometric congruence’s. First of all what do we know about all the outer or excircles? They all pass through point H by their construction so XH is a radius, ZH is a radius, and YH is a radius. But we know that by our construction points X and Y are one a line that is a perpendicular bisector of AH that crosses AH at point E. So XE is congruent to EY. And the two triangles share a side of EH as well as a ninety degree angle at E so by the Side Angle Side axiom XH is congruent to HY. And since YZ is on the same line which is a perpendicular bisector of HC through D then YD and ZD are congruent. And if we go through the same logic as before we will come to another SAS axiom that HY and HZ are congruent so all of the radii of our three outer circumcircles are the same. And since we know that X, Y, and Z are just reflections of the orthocenter of our original triangle we can say that its radius is also congruent to the other three. So our assumption that all the circumcircles in our orthocentric system are the same is true.

As promised here is some history of the Nine Point Circle. History

And here are three nice proofs that I found on the internet. Proofs

Write Up by David Drew
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