Area of the equilateral triangle:

Given : D, E, F are the mind points of BC, CA, AB.
R.T.P. : We have to determine the ratio of the area of of triangle DEF and triangle ABC.
Proof : F is the mid point of AB.
D is the mid point of BC.

FD||AC => FD||AE 

Similarly ED||AF
FE||BD
 AFDE is a parallelogram.
∠A = ∠D (Opposite angles in a parallelogram)

Similarly BDEF is a parallelogram.
∠B = ∠E

In ΔABC, ΔDEF

∠A =∠D proved

∠B =∠E proved

ΔABC ~ ΔDEF (A. A similarity)

 

Ratio of the areas of triangle DEF and triangle ABC = 1 : 4