First, let us consider a simpler problem:
Let ABC be a triangle, and let its medians be segments j, k, and m.
Start with segment k. Label its endpoints A and D. A and D are two vertices of the desired triangle. To find the other vertex, construct a circle having radius j and center A. Construct a second circle having radius m and center D. Construct the points of intersection of these two circles, and label them Y and Z.
We can construct two triangles having sides with the lengths of the three medians of the original triangle using segment k as one of the segments -- triangles ADZ and ADY.
The triangle that we want to consider, though, is triangle ADZ.
Notice that the segment ZA and segment j are parallel. Segment ZD and segment m are also parallel. We will use this when we construct a triangle given its medians.
Now that we have constructed a triangle with the sides having the lengths of the medians of a given triangle, let us return to the problem of constructing a triangle given its medians.
Suppose we are given three line segments: j, k, and m. If these are the medians of a triangle, then we want to construct the triangle
Let us begin by making a triangle having sides of length j, k, and m. Start with segment k. Label its endpoints A and D. Construct a circle having radius j and center A. Construct a second circle having radius m and center D. Construct the intersection of these two circles, and label it Z. ADZ is a triangle having sides of length j, k, and m.
We saw from the problem above that if segment k is one of the medians of the original triangle (the triangle we are trying to construct), then one of the medians is parallel to segment j, and the other median is parallel to segment m.
Point A will be one of the vertices of the desired triangle. We need to construct the other two vertices -- B and C.
We also know that the medians intersect at a point that divides each median in a 2:1 ratio. We need to find this point on segment k. We can do this by trisecting segment k (see script -- write-up 5). This point is the centroid of the triangle that we are trying to construct.
Now construct a line parallel to segment j through the centroid. Label the line l. 2/3 of segment j will be on one side of segment k, and 1/3 will be on the other side.
Therefore, we need to trisect segment j. Construct the circle centered at the centroid having a radius of 2/3 the length of segment j. The intersection of this circle with the line l forms one of the vertices of the triangle. Label this point B.
Construct the circle centered at the centroid having radius of 1/3 the length of segment j. The intersection of this circle and the line l is the midpoint of one of the sides of the triangle. Label this point E.
Since E is the midpoint of one of the sides (AC) of the desired triangle, then AE = EC. Therefore, we need to construct the point on the ray AE that is the same distance from E that A is. To do this, construct a circle centered at E through the point A. The intersection of this circle and the line AE gives us the point C.
The points A, B, and C are the vertices of the desired triangle.
We can construct the medians of triangle ABC, and we can see that they are the same as the given segments j, k, and m.
