Using Medians to Construct a Triangle

By: Sydney Roberts

For this exploration, we will show that we can construct a unique triangle given three line segments j, k, and l that represent the medians of the triangle. Segments j, k, and l are shown below.

Since these segments are medians, and we know that the medians of a triangle intersect at a point two-thirds of the distance away from the vertex. Therefore, we should start by trisecting each segment, so which have the point along each segment where the intersection should occur.

We know that each of these segments should intersect at the same point, but at what angle should the segments be intersecting so that they form the medians of a triangle? Well to find this we need to start by choosing one segment, say l, and label the point one-thirds of the way down the segment as C which stands for our centroid.

Then we want to find a point P that is one-third the length of the median, but going away from the centroid.

Now, we want to create a circle that is centered at P and has a radius that is two-thirds the length of segment k.

Now we want to implement the last segment, so we want to construct a circle centered at C, our centroid, that has a radius that is 2/3 the length of median j.

Where these two triangles intersect, we can label it as X, and it will end up being one of the vertices of our triangle. Then, since our red triangle has a radius that is two-thirds the length of j, then we can construct our median j by adding half of the length of our radius on the red circle in the opposite direction. (i.e. the radius is 2/3j and if we add ½ of this length, we are essentially adding the other 1/3 back, and hence we have all of j).

Now that we have one of the medians (segment j) exactly where we want it, we need to construct the other ones so they are where we want them. To do this, we want to basically repeat the process. So now we want a circle centered at XÕ that has a radius that is two-thirds the length of l.

And now we should add one more circle that is centered at C and has a radius that is two-thirds the length of k. Where these two new triangles intersect, we will label it as point Y and this will be the second vertex of our triangle.

Notice that we can no construct median k, because the length of segment YC is two-thirds of k. So we can construct this median the same way we constructed j, and add half of the length of segment YC to the opposite side of C as pictured below.

Cleaning it up so that we only see the medians, we get the following image.

Connecting X, Y, and the endpoint of our median l, we get the triangle we have been searching for. We can refer to this as triangle XYZ.

This triangle is unique in the sense that any other triangle with these same medians would be congruent to triangle XYZ . This triangle was constructed by starting with one of the given segments, and using circles to measure out distances that in turn were used to construct the other given segments.  If we had chosen different circles, or intersection points, we would not have been able to form a triangle. Hence these particular segments give us this particular triangle XYZ.