Today I am going to explore problem 2 on Jim Wilson's Assignment 6. During this exploration I plan to take a triangle, construct itís medians, and construct another triangle whose sides are the lengths of the medians of the first triangle. I plan to do this several times with different types of triangles and discover whether or not the two triangles have the same shape. For example, when the original triangle is an equilateral triangle, I want to know if the medians triangle will be an equilateral triangle. I also plan to explore an isosceles triangle and a right triangle. After I find an answer, I plan to figure out why that answer is true.

First I want to look at an equilateral triangle. From exploration I have discovered that both the original triangle and the medians triangle are equilateral.

Also from exploration I have discovered that a triangle constructed so that the sides are the lengths of the medians of an isosceles triangle is also isosceles.

Now from exploring a right triangle I have discovered that the medians triangle of the right triangle is not always a right triangle. I have discovered that the medians triangle is a right triangle only when two of the medians of the original triangle are perpendicular to each other.

But why?

By comparing the angles formed by the medians of an equilateral triangle, an isosceles triangle, a right triangle, and a triangle free to take any kind of triangular shape with each of their medians triangles we see that each angle of the medians triangle is equal to two of the angles formed by the medians of each original triangle. So for the right triangleís medians triangle to be a right triangle then two of the medians must form 90 degree angles so that the angle of the medians triangle that corresponds to the 90 degree angles will be 90 degrees. Hence, the right triangle.

But why are the angles formed from the medians of the original triangle equal to the angles of the medians triangle?