Three Medians, One Triangle

by: Al Byrnes

Part 1: You are given three line segments j, k, and m. Show that your construction is correct.

If these are medians of a triangle, construct the triangle.

First consider the three segments j, k, and m:

Using these three segments lets construct a triangle that will serve as the triangle for medians for our larger triangle:

Great! Let's construct the larger triangle by translating the medians of the triangles along vector lengths, whose magnitudes are the lengths of the other two medians of the larger triangle. So, to construct the first side of the triangle, we will use the mark vector function in GSP to translate segment j to the end of segment m:

Note how segment j was translated the length of segment m (direction of the translation is indicated by the arrow). To create the other half of the side of the triangle, we will make a similar translation move, however, along the segment k and in the direction indicated by the black arrows in the illustration below:

By the nature of the construction, we can be sure that the segments m and k coincide with the midpoint of the segment of length 2j (the length of each of the segments to either side of the point at which segments k and m conincide with the side of length 2j are both j, respectively.

To form the other two sides of the triangle, continue by similarly translating the segment of length k along the segment of length j and then a magnitute of m in the direction indicated by the arrows. See both steps for creating the second side of the triangle illustrated below:

Again, we can be sure that segements j and m coincide with the segement of length 2k at the segment's midpoint because the length of the segments on either side of the point of coincidence are half the length of the segment that forms the second. For the final side of length 2m, translate the median of length m (indicated in with the red coloration) along the segments of length k and the segment of length j as indicated in the illustration below:

So, we have constructed a triangle using the median segements j, k, and m. The side lengths of the triangle are 2j, 2k, and 2m, respectively.

Part 2: Show that the triangle is unique.

Proof by contradiction: assume that the triangles 2m2k2j is the same triangle as 2m2k2n and that the value of j and n are not equal.

These triangles can be constructed by their medians as illustrated in part 1. Here is an illustration of the two triangles and their medians, segments m, k, and j, and segments m, k, and n, respectively:

In order for the side lengths 2n and 2j to be equivalent, the length of the median segments must be the same, as each side length of the triangle is double the length of the median length by the nature of constructing the triangle using its medians. In this case, the only way that the sides of the triangles will be equivalent is if the lengths of the triangles' medians are equivalent. Therefore, since it was assumed that n ≠ j, 2n ≠ 2j and the triangle 2m2k2j constructed using the medians m, k, and j is unique.