Since P is the centroid of triangle ABC,
line segments AD, CF and BE are the medians of triangle ABC and
D, E and F are the midpoints of sides BC, AC and AB respectively.
Thus, BD = DC (call this length a), CE = EA (call this length
b) and AF = FB (call this length c). In other words, AB = 2c,
BC = 2a and AC = 2b. The **Triangle
Mid-Segment Theorem** states: In any triangle, a segment
joining the midpoints of any two sides will be parallel to the
third side and half its length. By this, FE = a, DF = b and FD
= c.

Using Heron's formula, the area of triangle DEF is

The semiperimeter of triangle ABC is

Using Heron's formula to find the area of triangle ABC:

Thus, when P is the centroid of triangle ABC, the area of triangle ABC is 4 times the area of triangle DEF.