Problem 2.
If the original triangle is equilateral, then the triangle of medians is equilateral. Will an isosceles original triangle generate an isosceles triangle of medians? Will a right triangle always generate a right triangle of medians? What if the medians triangle is a right triangle? Under what conditions will the original triangle and medians triangle both be right triangles?
First, let's look at the triangle of medians for a couple equilateral triangles. (The median triangle is in yellow)
Looking at these constructions we see the triangle of medians for an equilateral triangle is also an equilateral triangle. The sides of the triangles of medians were also measured in GSP to verify that the triangles of medians are equilateral. From these constructions and similar constructions done in GSP we can conclude that the triangle of medians for an equilateral triangle will always be an equilateral triangle.
Next, we will look at the triangle of medians for a couple of isosceles triangles. (The median triangle is in yellow)
Looking at these constructions we see that triangle of medians for both constructions is also an isosceles triangle (side lengths measured in GSP to confirm). From these constructions and similar constructions created in GSP we can conclude that the triangle of medians of an isosceles triangle will always be isosceles.
Next, we will look at the triangle of medians for a couple of right triangles.
Looking at these constructions the triangles of medians appear to be right triangles, but when the angles are measured it is discovered they are close but not actually right triangles.
Next, let's look at the triangle of medians for a couple of isosceles right triangles.
Looking at these constructions the triangles of medians appear to be acute, but not right triangles for an isosceles right triangle.
Next, we will look at right triangles that also have triangle of medians that are right triangles.
Since the original triangles are right triangles, one of the angles is 90 degrees. We notice when both the original triangle and the triangle of medians are right triangles the other two angles measure 35 degrees and 55 degrees.