A Lesson on the Nine Point Circle and the Euler Line

Introduction
 

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.  The nine-point circle was discovered by Karl Wilhelm Reurbach (1800-1834).  The midpoints of the sides of a triangle, the points of intersection of the altitudes and the sides, and the midpoints of the segments joining the orthocenter and the vertices of a triangle all lie on the nine-point circle.

The Euler line is the line containing the orthocenter, centroid, and circumcenter, named after Leonhard Euler (1707-1783).
 
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Activity 1
 

To understand this lesson, one need to be familiar with the three concurrency points in a triangle and the Euler Line.

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Activity 2
 

Directions:  Below are eight questions concerning the nine-point circle and Euler line, click on Figure 1-3 to download a sketchpad sketch of the given figure.  Use Figure 1 to answer questions 1 and 2, Figure 2 to answer questions 3 and 4, and Figure 3 to answer questions 5-8.  (Downloadable Geometer's Sketchpad - Demo Version)
 
Figure 1 (use for questions 1 and 2)
 

1.  Using the GSP's arrow icon, drag the vertices around and observe the effect on the nine-point circle.  Is it possible to position the vertices so that each side is intersected only once by the circle?  If so, under what circumstances does this occur?  If not, why not?

2.  When the triangle is isosceles, what happens to the nine points?
 

Figure 2 (use for questions 3 and 4)
 

3.  State a conjecture concerning the positions of the orthocenter, centroid, and circumcenter.  To prove your conjecture false, what sort of counter example would be necessary?

4.  State a conjecture concerning the following ratios:  (centroid-center of the nine point circle)/(centroid-circumcenter) and (orthocenter-center of the nine-point circle)/(orthocenter-circumcenter).  To prove your conjecture false, what sort of counter example would be necessary?
 

Figure 3 (use for questions 5 - 8)
 

5.  State a conjecture concerning the relative positions of the orthocenter, circumcenter, and center of the nine-point circle.  To prove your conjecture false, what sort of counter example would be necessary?

6.  Is it possible to position the vertices so that the orthocenter, circumcenter, and center of the nine-point circle coincide?  Is so, under what circumstances?  If not, why not?

7.  State a conjecture concerning the radius of the circumcirle and the radius of the nine-point circle.  To prove your conjecture false, what sort of counter example would be necessary.

8.  State a conjecture concerning the manner in which the nine-point circle divides segments drawn from the orthocenter to points on the circumcircle.  To prove your conjecture false, what sort of counter example would be necessary?

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Follow - Up Activity
 

9.  What did you like and dislike about this lesson?  

10. Would you consider Geometer's sketchpad as a helpful tool for developing geometric models?  How helpful was Geometer's Sketchpad to developing your own conjectures?  Explain.

11. Give a previous theorem or geometric property about an object that we learned about early that would have been beneficial to use Geometer's Sketchpad to explore it's properties.  (Explain.)  If you feel there were none tell why you feel this way.  

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