In the following explorations we look at isoceles, equilateral, and right triangles. My exploration is to see if when I construct the median triangle will it always be the same as the original triangle. I.E. that if I constructed an equilateral triangle with its median triangle be an equilateral triangle. This is demonstrated in a GSP sketch.
Click here to see GSP sketches.
Equilateral Triangle: An equilateral triangle will always have an equilateral median triangle. The reason for this is because all of the sides are of equal length and the median triangle is half the size of the original lengths. So if my original lengths were 4 then my median triangle would have lengths of 2. The original triangle is basically dilated by one half and then rotated to form the median triangle. Therefore, this is why an equilateral triangle formed by the medians will create another equilateral. On top of that it is also because a dilation will preserve angle measurements.
Isoceles Triangle:An isoceles triangle will also create a triangle of medians that is isoceles for the same reasons discussed above.
Right Triangle: At first I constructed a right triangle and was trying to explain why the median triangle is a right triangle. But I remembered that just because I have right angle does not necessarily mean that it is a right triangle. My first picture shows a right triangle with lengths 3,4,5. I then did an example of a right triangle with angles of 30,60,90 degrees, but the lengths of the sides did not work for a right triangle. As mentioned above my original triangle again is dilated by .5. As far as the orientation of the triangle it is rotated so that the midpoint opposite the right angle is the right angle for the median triangle.
Conclusion: Therefore, If you were to create the median triangles for an isoceles, equilateral, and right triangle(with appropriate lengths) you will always get a median triangle that is isoceles, equilateral, and right.