**Given triangle ABC with medians CH, BF,
and AG, we construct a triangle with the three sides having the
lengths BF, CH, and AG. Now, we want to explore the relationship
between the original triangle and the new one.**

**First, we find the perimeter and area of
each triangle. Then, calculate the ratio of the new triangle to
the original triangle in for both perimeter and area.**

**Notice that the ratio of the perimeters
is not included, instead we see the ratio of the perimeters squared.
Comparing this ratio to the ratio of the areas, we find that the
ratios are equal. Will this relationship hold for any triangle
and the triangle with its three sides the same length as the medians
of the origianl triangle? Click here
to try.**

**You will notice as you manipulate the original
triangle, the ratio of the area remains constant at 0.75. Can
you find instances where the ratio of the perimeters squared is
not 0.75? The question to investigate becomes when is the ratio
of the perimeters squared not 0.75?**

**If you would like to start with your own
triangle, click here to run the
script used in the example.**