An Investigation of
the 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.

Start by constructing the altitudes of the triangle and the orthocenter. The points where the altitudes intersect the sides of the triangle are called the feet of the altitudes. What do you notice about the feet of the altitudes? (Check your answer.)

Next construct all three angle bisectors in your triangle. You may want to change your line color (Display Menu) so you don't get confused. Do the angle bisectors, the circle, and the triangle all seem to intersect in a point? Be sure to drag the vertices of your triangle to check your conjecture. (Check your answer.

We still have important segments like medians and perpendicular bisectors of sides to try. Since both of these segments pass through the midpoints of the sides, try constructing these midpoints. Are these points on the circle? (Check your answer.)

The last three points are tricky to find. They lie on the altitudes and are called Euler points. To find them, construct a segment from a vertex to the orthocenter and find it's midpoint. If your construction is correct, this point should lie on the circle. Finish constructing the other two Euler points.

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:

Constructing a circle given three points on it.
Finding three points on the circle to use for the construction.

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

Constructing the Circle with Transformations

The Nine Point Circle has many amazing properties that continue to fascinate mathematicians and math students alike. One of the properties is that it can be constructed from another circle using a mathematical transformation called a dilation. If you have never tried to dilate an object using Sketchpad, work through the mini-investigation about dilation before you continue.

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.

Both the orthocenter and the circumcenter lie on Euler's Line and the center of the nine point circle is related to these two points by dilation. Make a conjecture about the relationship between the nine point center and Euler's Line. You can check your conjecture by clicking on the Show Euler's Line button on the sketch.

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

On your sketch, construct a segment from the orthocenter to the nine point center and measure it's length.

Construct and measure another segment from the circumcenter to the nine point center.

What do you notice about these two segments? Move your triangle around to test your conjecture. (Check your answer.)

Can you name four points on Euler's Line? (Check your answer.)

Before you leave this investigation....

Did you follow your teacher's directions about saving your work and turning it in?

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