The medians of a triangle are segments from each vertex to the mid-points of the opposite sides.
Our goal is to construct a second triangle with the three sides having the lengths of the three medians from the first triangle.
This triangle was constructed by selecting a point, not on triangle ABC, and constructing a circle by using this point as the center and the line segment, o, as the radius. Then, after choosing a point on this circle, construct the line segment joining the two points. This creates a line segment of length o, which we'll call o'. This will be the base of our new triangle. To construct the next side, select an end point of this segment, then construct a circle with this point as the center and segment n as the radius. Then choose the other end point of segment o', construct another circle with this point as the center and segment m as the radius. After constructing the two circles choose a point of intersection of the two circles. This point will be a vertex of the new triangle. We now need to construct the line segments connecting the end points of o' to this new vertex. This will form the other two sides of the triangle, m', and n'. See picture of construction below.
To see if the sides of the new triangle are the same lengths of the medians of the original triangle, I had GSP calculate the lengths:
Notice, the lengths o', n', and m' are the same lengths as the medians of the original triangle, o, n, and m, respectively.
The next goal is to see if there is any relationship between the triangle of medians and the original triangle. Are they congruent? Using GSP to calculate the lengths of the sides of triangle ABC and the length of the sides of the triangle of medians, we get,
It is clear that the two triangles are not congruent, since the length of their sides are not equal. Perhaps the two triangles are similar. Using GSP to calculate the measure of their angles of the original triangle and the triangle of medians respectively, we get
Since none of their angle measurements are the same, we can see that the triangles are not similar. Do they have the same perimeter?
As we can see, the perimeter of the original triangle is 7.763 inches, while the perimeter of the triangle of medians is 6.739. Although the perimeters are relatively close, they are not the same. Lets look at the area of the two triangles.
Clearly, the two areas are not the same. However, there may be a relationship between their areas. The ratio of the area of the triangle of medians to the area of the original triangle is
This is a rational number, 3/4. Does this occur for all triangles constructed from the original triangle's medians? Do the perimeters of the triangles have a similar relationship? The ratio of the perimeter of the triangle of medians and the original triangle is
Is this always the case? Let's construct another triangle of medians and look at the ratio of their perimeters and areas. Use the script to perform this construction. Then move a vertex of the original triangle around. Note: You must have Geometers Sketch Pad installed on your computer in order to run the script. Notice that the ratio of perimeters changes as the point is moved around, but the ratio of areas remains fixed. This makes a very convincing argument that the ratio of the area of the triangle of medians to the area of the original triangle is always 3/4. The next goal is to prove that this is always the case.
It is known that the length of a median can be found from the lengths of the three original sides of the triangle. If the lengths of the sides of the original triangle are length a, b, and c, then the length of the median from vertex A to side a is given by
The area of the original triangle ABC, using Heron's Formula, is
The area of the triangle of medians is
The rest of the proof is done using MAPLE to perform the tedious algebra.
Now we can see that for all triangles constructed from the medians of the original triangle, the ratio of the area of the triangle of medians to the area of the original triangle is 3/4.