__THE QUESTION?__

If you have an equilateral triangle and create a new triangle of the medians - will the new median triangle be equilateral also?

Look at the figure below created using GSP. The original triangle is ABC and the three medians are Cf, Ad, and Be. I chose to use the median Be for the base of the new median triangle. Then, in order to finish constructing this new median triangle, I used GSP's parallel line construction tool . This tool created the 2 lines eG and BG and as you can see these two lines intersect at point G - and the new median triangle is formed (shaded in green). Because ABC is an equilateral triangle, the 3 medians are also equilateral and equal to the three sides as well. Thus the new median triangle is an equilateral triangle just as the original.

__THE QUESTION?__

If you have an isoceles triangle and create a new triangle of the medians - will the new median triangle be isoceles also?

Using GSP, I drew the isoceles triangle ABC (using the perpendicular line tool to create an altitude for the triangle). The three medians are Ae, Cd, and Bf. Again I selected a base, Ae, for the median triangle. Then I used the parallel line tool in GSP to create the other 2 sides of the new median triangle, AG & Ge. As shown in the diagram below, the new triangle AGe is also an isoceles triangle (shaded in blue) - notice the measurements of the line segments' lengths.

__THE QUESTION?__

If you have an right triangle and create a new triangle of the medians - will the new median triangle be a right triangle also? If so, in what instances will the median triangle also be a right triangle?

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When is the median triangle a right triangle? As you can see in the diagram below, when the original right triangles' measure 90, 55, and 35 degrees, the median triangle formed will be a right triangle.