Write-Up #4--Euler's Line


The use of technology in geometry classes allows teacher's to investigate more ideas in various subjects. If teacher must wait on the students to complete constructions during class time or even at home, there is a lot of time that is wasted. The students will also get extremely tired of doing many constructions demonstrating the same concept for the three types of triangles (acute, obtuse, and right). Therefore, the use of any geometry software package in a class will decrease the time spent on the "grunt work" and leave more time for the "ah-ha" feeling that students get when they discover an idea by using the software.

One example of such an idea is in the discover of Euler's Line of a Triangle. Euler discovered that the centroid, the orthocenter, and the circumcenter of a triangle lie on the same line. This line is now named for him. By using a geometry package, (Geometer's Sketchpad, in this case) the students can find the centers of the triangle and explore their movement as the shape of the triangle changes.



Click on the triangle to see an animation of the triangle on Geometer's Sketchpad.

Once the students create the triangle with the centers, they can manipulate the triangle to see what happens with the different types of triangles. They can also see what happens in very special cases, like a right triangle.

Directions for creating Euler's Line:

1. Create any triangle.
2. Construct the centroid (G) of the triangle. A centroid is the intersection of the medians of the
triangle. A median is the segment from a vertex to the midpoint of the opposite side.
3. Construct the orthocenter (H) of the triangle. The orthocenter is the intersection of the altitudes of the triangle. An altitudes is the perpendicular segment from a vertex to the opposite side.
4. Construct the circumcenter (C) of the triangle. The circumcenter is equidistant from the three vertices of the triangle. It lies on the perpendicular bisector.
5. Construct the incenter (I) of the triangle. The incenter lies on the intersection of the angle bisectors of the vertices.
6. Draw a line from the orthocenter to the circumcenter.
If you are interested in the Geometer's Sketchpad script, click here.

First, try some manipulations yourself. Would you like to see some examples with specific triangles: acute triangles, obtuse triangles, and right triangles?


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