A median of a triangle is defined as a segment connecting the vertex of a triangle with the midpoint of the opposite side.

If three arbitrary segments were given, it would be fairly easy to construct a triangle of those given lengths and the three medians of the triangle. However, if the three medians were given of a triangle first, could the triangle they came from be constructed? This construction can be made with the use of parallelograms and the trisect ion of a segment.

The following GSP sketch illustrates the given medians and the resulting triangle that has the given medians.

**Click here** to manipulate the GSP sketch to different median lengths.

In order to get further insight into the construction, it might be helpful to show two key steps:

1) Use the given medians to construct a triangle. Then use one of the sides as a median of the triangle you are about to construct.

2) The median you choose as one inside the triangle needing construction requires trisection. Then look for parallels with the other medians to complete the triangle.

The following shows the construction steps of the illustration above without hidden objects:

Note the trisection of a segment (the circles). It may
be helpful to work the construction backwards and then proceed with step
by step construction. **Click here for a
GSP script** that will do the construction of the triangle from the
medians. You will need to highlight three triangle vertices and two other
arbitrary points used for trisection purposes (5 points highlighted in all).