Points, lines and circles associated with a triangle

Lernpfad erstellt und betreut von:

Sandra Fink

E-mail: sandra.fink@edu.uni-graz.at
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Übersicht:       
 Hilfe
1. Elementary Geometry Revisited
2. Euler Line
3. Nine-Point-Circle

Nine-Point-Circle
 
3.1 Introduction

In this section we will investigate nine specific points of a triangle. Moreover, these points have the surprising relationship of all being on the same circle. This circle is called the nine-point circle of the triangle.

Click here to "check" this! You will be able to change the size of the triangle by grabbing vertex A, B or C and see that these nine special points still lie on one circle.
activity, take notes in your portfolio
 
3.2 Nine-Point Circle

Definition:
The nine-point circle is a circle that can be constructed for any given triangle. It is so named because it passes through nine significant points of the triangle. These points are:

  • the midpoints of the sides
  • the feet of the altitudes
  • the midpoints of the segments from the orthocenter to the vertices.

I.e. these points are concyclic.

The nine-point circle is also known as Feuerbach's circle or Euler's circle.


A little bit of history...

In 1765, Leonhard Euler showed that six of these points, the midpoints of the sides and the feet of the altitudes, determine a unique circle. Yet, not until 1820, when a paper published by Charles-Julian Brianchon and Jean-Victor Poncelet appeared, were the remaining three points (the midpoints of the segments from the orthocenter to the vertices) found to be on this circle. Their paper contains the first complete proof of the theorem and uses the name "the nine-point circle" for the first time.


take notes in your portfolio (+ a sketch)
 
3.3 Exercises

Exercise 1:
Try to show with GeoGebra that the radius of a triangle's circumcircle (i.e. circumradius) is twice the radius of that triangle's nine-point circle.

Exercise 2:
Show that the center of any nine-point circle (i.e. the nine-point center) lies on the triangle's Euler line. In fact, it lies at the midpoint between that triangle's orthocenter and circumcenter.


exercise
 
3.4 Nine-Point-Center
It should be clear from the application above that the orthocenter H is not the center of the nine-point circle. In fact the center of the nine-point circle is a new triangle center that we have not encountered before.

Definition:
The center of the nine-point circle is the nine-point center of a triangle ABC. It is denoted by N. The nine-point center is the midpoint of the line segment from the circumcenter to the orthocenter.


Exercise 3:
Try to prove the nine-point center theorem with the help of GeoGebra.

Exercise 4:
Construct the nine-point circle for any triangle ABC without the help of a computer software. The following link may be helpful: how to construct a circle through 3 points


take notes in your portfolio, exercise
 
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