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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1. Elementary Geometry Revisited
2. Euler Line
3. Nine-Point-Circle

Euler Line
2.1 Introduction

In this section we will investigate the relationship between the triangle properties that were mentioned in the previous chapter "Elementary Geometry Revisited". The theorem was discovered by the German mathematician Leonhard Euler (1707–1783).

Construct a triangle with vertices A, B and C with GeoGebra and construct all three centers G, H and O. Hide any lines that were used in the construction so that only the triangle and the three centers are visible. Put a line through two of the centers and observe that the third also lies on that line. Verify that G, H, and O continue to be collinear even when the shape of the triangle is changed.

Don't forget to save your file!!

2.2 Euler Line Theorem

The line through H, O, and G is called the Euler line of the triangle.

In the exercise above you should have discovered the following theorem:

Euler Line Theorem:
The orthocenter H, the circumcenter O, and the centroid G of any triangle are collinear, i.e. they all lie on one line. Furthermore, G is between H and O (unless the triangle is equilateral, in which case the three points coincide) and HG = 2GO. I.e. the centroid of a triangle trisects the segment from from the orthocenter to the circumcenter.

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2.3 Euler Line Proof

Euler Line Theorem - PROOF:

A pretty easy Proof of the Euler Line Theorem can be found here. Try to follow the steps in GeoGebra.
Don't forget to save your file!

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2.4 Exercises

Exercise 1:
Show (with the help of GeoGebra) that a triangle is equilateral if and only if its centroid, orthocenter and circumcenter are the same point. In case the triangle is equilateral, the centroid, the orthocenter, and the circumcenter all lie on one point.

Exercise 2:
Let's find the point P within a triangle ABC, so that if we draw lines from P to the three vertices, we get three equal (area) triangles. Is this P perhaps a different center of the triangle, from those mentioned in chapter 1?

Exercise 3:
In triangle ABC, the angle bisector of A meets the altitude from B, and the median from C in one point P. Does the triangle have to be equilateral? If not, give an example that is not an equilateral triangle.

Exercise 4:
Show (with the help of GeoGebra) that the circumcenter of a given triangle is the orthocenter of the triangle formed by joining the midpoints of the sides of the original triangle ABC.

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