Assignment 4: Playing with Euler

By Krista Floer

Take any triangle. Construct a triangle connecting the three midpoints of the sides. This is called the MEDIAL triangle. It is similar to the original triangle and one-fourth of its area. Construct G, H, C, and I for this new triangle. Compare to G, H, C, and I in the original triangle.

First, let's look at a construction of the 4 centers in the large triangle. Click HERE for the GSP file.

We see that G, H, and C all lie on a line. This is called the Euler line. In this case, the incenter does not lie on the line. Now we will look at the same triangle with the medial triangle constructed also. The centers of the medial triangle are labeled.

We can see that the circumcenter, centroid, and the orthocenter of both triangles all lie on the same line. The letters that are underlined are the centers of the large triangle, and the ones that are not are for the medial triangle. We see that the circumcenter of the large triangle corresponds to the orthocenter of the medial triangle. The centroid is the same for both triangles.

Click HERE for the GSP file to investigate for yourself.

Something interesting that I found while playing around with the sketch was that the Euler line for the medial triangle seemed to always have the length of the Euler line for the large triangle. The proof that I came up with is this:

We want to show that the Euler line for the medial triangle is half the length of the Euler line for the large triangle.

Proof:

We are given a triangle. Then we construct the medial triangle. From the midsegment theorem, we know that each leg of the medial triangle is half the length of the leg it is parallel to. From this we see that the proportion of the medial triangle to the original triangle is .5. Hence, the the Euler line of the medial triangle is half the length of the Euler line for the original triangle.