In this assignment, I studied two triangles, one of which was constructed using the medians of the other, here is a picture of what I am talking about:

Triangle AGL was constructed from the medians of triangle ABC. After doing the construction, I then looked at some measurements to see if their were any relationships between the two triangles, here are the measurements:

Perimeter(Polygon BAC) = 44.68 cm

Area(Polygon BAC) = 84.84 square cm

Length(Segment AG) = 9.37 cm

Length(Segment EC) = 13.71 cm

Length(Segment BF) = 15.50 cm

Perimeter(Polygon GAL) = 38.58 cm

Area(Polygon GAL) = 63.63 square cm

Length(Segment GL) = 15.50 cm

Length(Segment AL) = 13.71 cm

Perimeter(Polygon GAL)/Perimeter(Polygon BAC) = 0.864

Area(Polygon GAL)/Area(Polygon BAC) = 0.750

From the given data, there appears to be a relationship between the areas of the two triangles, that is, the area of the median triangle to the area of the original triangle is in a ratio of 3:4. I then did a similar construction and scripted it and this is what I found:

Area(Polygon ABC) = 58.81 square cm

Length(Segment AF) = 7.80 cm

Length(Segment BG) = 12.33 cm

Length(Segment CE) = 11.66 cm

Length(Segment AF) = 7.80 cm

Area(Polygon AFL) = 44.11 square cm

Length(Segment FL) = 12.33 cm

Length(Segment AL) = 11.66 cm

Area(Polygon AFL)/Area(Polygon ABC) = 0.750

The ratio of 3:4 between the two areas holds true for this case as well making it seem that this is a valid ratio relationship between any triangle and it's median triangle. To see the construction of the second display click on the following:

construction of triangle and it's median triangle

I am not exactly sure of the proof of this but have made two more scripts of two different constructions to help support my hypothesis: