Exploration of triangles and their medians

By: Moni Olubuyide

1. Construct a triangle and its medians.

Construct a second triangle with the three sides having the lengths of the three medians from your first triangle.

Find some relationship between the two triangles. (E.g., are they congruent? similar? have same area? same perimeter? ratio of areas? ratio or perimeters?) Prove whatever you find.

As you can see the two triangles are neither congruent or similar because they have different angles and sides that are unproportional.

Moreover, they don't have the same area or perimeter. At this point, you would want to say that there is no relationship between these two triangles but there is one that I found => The ratio of the area of the two triangles is ALWAYS 4:3 or 1.3. The ratio of the perimeter although is not constant is ALWAYS between 1.0 and 1.3.

CLICK HERE TO SEE ANIMATION of ratios

2. If the original triangle is equilateral, then the triangle of medians is equilateral.

Will an isosceles original triangle generate and isosceles triangle of medians? YES

Will a right triangle always generate a right triangle of medians? YES and NO

The medians of a right triangle can form a right triangle but if you click here you can see that this is not always true.

What if the medians triangle is a right triangle?

Here Triangle FGH is the median triangle. Then I constructed triangle HJK, whose medians are the lengths of triangle FGH. If you click here for the animation you can see that in this case also the original triangle is not necessarily a right triangle.

Investigate for yourself under what conditions will the original triangle and the medians triangle both be right triangles?