Given line segments a,b,c. If these are the medians of a triangle, construct the triangle. Now what information do we have about medians. We know that they are the segments going from the vertices of a triangle to the middle of the opposite side. They meet in a common point called the centroid. The distance from the vertex to the centroid is twice the distance from the centroid to the middle of the opposite side. Note that these segments are divided into thirds. This can be done with GSP , either by using a script or by construction.
Now let's form a triangle using these medians as the sides of the triangle. First we copy c. Then using the left endpoint of c as centerpoint and b as a radius, construct a circle. Now using the right end of c as centerpoint and a as a radius construct a circle. Note that these circles correspond to the arcs used when constructing a triangle from given segments with compass and straightedge.
Connect the ends of segment c to the intersection of the two circles. These segments are the lengths of a and b.
Now we have a triangle made from the medians, but that isn't what we want in the end. Let's hide the circles to get them out of the way and then construct a line parallel to a through the centroid G.
Now on the line parallel to a we'll mark off 1/3 on one side of the centroid and 2/3 on the other side by using G for the center and getting the radii from segment a.
Let's keep the segment we've just constructed , but hide the rest of the line and the circles.
We'll repeat that procedure by drawing a line through the centroid parallel to b, then marking off 1/3 and 2/3 of it as above.
Now we have the three vertices of the triangle we're looking for. Let's connect the endpoints of each median that has the 2/3 section to form the triangle. Remember that the red lines are the original medians, the blue lines are copies of medians a and b, and the yellow lines are the sides of the triangle made from the medians.
In making the triangle we used the following properties of medians: they meet at a point called the centroid,and the centroid divides the medians into 1:2 ratio with the longer piece connecting the vertex and the centroid. But the definition of median also includes the fact that it is connected to the midpoint of the opposite side. In the above drawing it certainly appears that each median connects to a mid-point of the new triangle, but let's measure to be sure.
The measurement of each 1/2 of a side is equal to the other 1/2. So we now have a triangle with the given medians.