Construction of a Triangle by its Medians

Colleen Foy

In the problem we are given three line segments j, k, and l. We are to suppose they are all medians of a triangle and to then construct the triangle.

So suppose we are given three line segments j, k, and l like so:

Now we want to construct a triangle with these segments:

Now we want to trisect each segment of the triangle:

We know that the intersections of the medians of a triangle is called the centroid and the centroid is located 2/3's along each median. Note that we only needed to trisect two of the segments since the third will follow. If we can get all three medians to intersect at one of the trisection points, we will have located the centroid of the unknown triangle. To do this we are going to copy each median and construct them to be concurrent at at point, let's say . We will also use the point .

Next, we will construct a line through parallel to segment AB and we construct the segment . Then we construct two circles with centers A and B and radius . Label the intersection of the circles with the parallel line through A' and B' respectively.

Now we have copied the segment AB to go through . Consequently we have two medians (AC and A'B') intersecting at 1/3 the length of the median. Next we will do the same for segment BC. First we construct a line through parallel to BC and construct the line segment . Then we construct two circles centered at B and C respectively with radius .

Label the intersection of the two circles with the parallel line to BC B'' and C' respectively. Now we have the third median (B"C') intersecting at a point 1/3 the distance of the median.

By definition, is the centroid of our original circle because it lies 1/3 the distance of each median. Now we construct the triangle with medians, B''C', A'B', and AC.

Now we have our original triangle in yellow. Click here for a GSP file to further investigate.