This write up explores and proves some facts about relationship between any given triangle and triangle formed by the medians of the given triangle.
Click here to open the GSP file and explore the relationship between the given triangle and triangle of its medians, the values such as angles, perimeter, area, ratio of perimeters, ratio of area for different types of given triangle by dragging one of the vertices of the triangle.
You observe through exploration, as you see in the picture below that the triangles are neither congruent nor similar looking at the angle measure. The perimeters or their ratios have no particular relationship. When we do look at the ratio of area of the median triangle to that of the given triangle, we notice that the ratio is a constant and equal to 0.75.
Lets try to prove this observation assuming some of the known facts about medians and centroid (point of concurrency of medians of a triangle) and area of a parallelogram. Click here for GSP file or look at the picture below.
We know that AG = (2/3) AE and so on CG = (2/3) CD and BG = (2/3)BF.
We also know that medians divide the triangle such that the area of the divided triangles are equal (or half the original triangle).
HereÕs the construction to help with our proof.
CGÕ is parallel to GB. DC is parallel to PBÕ and GP is perpendicular to DC.
Consider the similar triangles CGGÕ and CDH.
We know CG = (2/3)CD. Hence by similar triangle areas property,
Area of CGGÕ = (2/3 *2/3) area of median triangle. = (4/9) area of median triangle.
But, from parallelogram, CGBGÕ, area of triangle GBGÕ = (4/9) area of median triangle.
And also, that area of triangle BGE = area of triangle EGC. Hence area of triangle GBC = (4/9) area of median triangle----------------------(1***)
Similarly constructing some more parallel lines we can prove that area of triangle EGC = area of triangle CGF = area of triangle AGF = area of triangle DGA = area of triangle BGD.
Hence area of triangle GBC = (1/3) area of triangle ABC -------------(2***)
From (1***) and (2***) we have
(4/9 ) area of median triangle = (1/3) area of triangle ABC
The ratio, area of the median triangle /area of triangle ABC = (3/4) = 0.75