Proof of the Ratios = 4

By:
Ginger Rhodes

**Given:** P is inside of triangle ABC, P is the centroid

**Prove:** the ratio of the areas of triangle ABC and triangle
DEF equals 4

First, to make it easier to discuss I will label AB = x, AC = y, and BC = z. Now, F, E, and D are the midpoints of the sides of the triangle, and therefore FE, FD, and DE are parallel to the respective side and are half the length of the sides. So FE = 1/2 z, FD = 1/2 y, and DE = 1/2 x.

Using Heron’s formula
the area of triangle ABC is

and the area of triangle FED
is

Now, Triangle ABC / Triangle
FED = 4