# An Investigation ofthe Nine Point Circle

## Background

The Nine Point Circle, named for the nine easily constructed points on it, has many exciting mathematical properties. Every triangle has a nine point circle which is connected to both it's inscribed circle, circumscribed circle, and Euler Line. It was first proven by French mathematicians Jean-Victor Poncelet and Charles Brianchon in 1821 and further explored by Karl Willhelm Feuerbach who discovered many of its properties.

In order to understand this exploration, you need to be familiar with the four concurrency points in a triangle and the Euler Line.

## An Exploration of the Nine Point Circle

We are going to start this investigation by finding the points on the Nine Point Circle before we try to construct it. Click on the sketch of the circle and follow the instructions below.

Checkpoint: Can you name the nine points on the Nine Point Circle (Check your answer.)

## The Euclidean Construction of the Circle

Now we are going to use some common properties of circles to construct the Nine Point Circle. We can construct any circle if we know three points on it, but we must be very clever. If you think you have a method click on the triangle sketch below and test your method. You can check it by constructing some of the points that should be on the circle. If you can construct the proper points on the circle and they stay there even when you move the triangle, your construction must be correct. After you try some of your ideas, you can check the hints below. Any time you need a new sketch to play with, just click on the picture and you will get another triangle.

Construction Hints:

Checkpoint: Did you check your circle to make sure it passes through the right nine points?

## Constructing the Circle with Transformations

Now we are going to construct the Nine Point Circle by dilating the circumcircle with the orthocenter as the center of dilation. Yes, these two centers of concurrency are connected. Before you start your construction, let's take a look at how the circumcircle dilates to become the Nine Point Circle. Click here to bring up the sketch.

At this point, you need to verify that this dilation actually works. You can click on the picture below to get a sketch with the circumcircle on it. Follow the directions below the picture to do the dilation.

## Construct and Investigate

• Select the Orthocenter and pull down the Transform Menu to Mark Center.
• Select the Circumcircle and pull down the Transform Menu to Dilate.
• Type in a ratio of 1/2. (1 = new, 2 = old) and click on Okay to dilate.
• Check to make sure your circle passes through the correct nine points by clicking on the Show button to show the points.

Now we need to find the center of the Nine Point Circle. It would make sense that if the Nine Point Circle was a dilation of the circumcircle in the orthocenter, that the center of the Nine Point Circle would be the same dilation of the circumcenter in the orthocenter.

• Use the Transform Menu to mark the orthocenter as the center of dilation.
• Select the circumcenter and dilate it by a ratio of 1/2.
• Check your construction by clicking on the Show Center button.

Next you will find the exact location of the nine point center.