Roxanne Kerry

I explored the following problem for my write-up number 6:

If the original triangle is equilateral, then the triangle of medians is equilateral. Will an isosceles original triangle generate an isosceles triangle of medians? Will a right triangle always generate a right triangle of medians? What if the medians triangle is a right triangle? Under what conditions will the original triangle and the medians triangle both be right triangles?

A median of a triangle, by definition, connects the midpoint of one side to the vertex of the opposing two sides in a triangle. For example, in the triangle ABC (shown below) we can find the midpoint of AB, label it c, and connect it to the vertex C to construct one of the medians of this triangle.

In order to make a triangle of medians as the problem calls for, you must first find the three medians of the triangle and use them to construct a new triangle. This will be your triangle of medians. The problem first states that the triangle of medians formed by an equilateral triangle is also equilateral (shown below), which makes sense since the equilateral triangle has sides of equal length, so the midpoints of each side will be the same distance from each opposing vertex. The angles shown below show that both triangles are equilateral (the angles of the medians triangle are estimated and slightly off, but they all have the same 60º angles if it was constructed accurately). The lengths of the three sides more accurately shows this same thing, that these are both equilateral triangles, since all three sides have the same lengths.

As for the triangle of medians constructed from an isosceles triangle, it turns out that this too forms an isosceles triangle. Using the same logic as was used for the equilateral triangle and its three sides of the same length, an isosceles triangle by definition has two sides of the same length, so the midpoints of the two isosceles sides will yield medians of the same length, thereby creating an isosceles triangle of medians, as shown below. Two of the angles in the original isosceles triangle are the same as is the case with the isosceles triangle of medians. You can also see that two of the sides in the original isosceles triangle have the same length, and two of the sides in the isosceles triangle of medians also do, confirming that triangle to be, in fact, isosceles as well.

As for the triangle of medians formed from a right triangle, I came to the conclusion that this triangle of medians does not, in general, produce a right triangle as the other special types of triangles seemed to do with their triangle of medians. The rationale for this is that with the equilateral and isosceles triangles, we are concerned primarily with the lengths of the sides of triangles, rather than the angles. When a triangle is equilateral, the sides all have the same length and it so happens that the angles are all the same, and with an isosceles triangle two of the sides have the same length and it so happens that two of the angles will also have the same length. However, a right triangle can be formed with far less restrictions on the lengths of sides. The primary concern in finding a right triangle is that there is a 90º, or right angle for one of the angles. If you notice in the above isosceles example, even though two of the angles were the same as each other in each of the two triangles, they were not the same angle measures when you compare the angle lengths of the original triangle to the triangle of medians. Since constructing a triangle of medians does not preserve angles, creating a triangle of medians from a right triangle will not typically form another right triangle. By the image below, we see that an isosceles right triangle does not yield a right triangle of medians.

Working the other way around, we can consider what kind of “parent triangle” would be made given a right triangle of medians. Just as in the above observations, this will not typically yield a right parent triangle. In my example (shown below), I made a generic right triangle of medians (not isosceles) and constructed its parent triangle by working backwards. I know that the place in which the three medians of any triangle cross is the centroid, which splits each median up into one third and two thirds. Using this knowledge, I split each of the three sides of the median triangle into thirds and put them together in such a way as to create the orthocenter, and then the original triangle. Although it is close to a right triangle, it is not quite a right triangle, so working backwards, a right triangle of medians does not typically come from a parent right triangle.