D, E and F are respectively the mid-points of the sides BC, CA and AB of a ∆ ABC. Show that 

(i) BDEF is a parallelogram. 

(ii) ar (DEF) = 1/ 4 ar (ABC) 

(iii) ar (BDEF) = 1 / 2 ar (ABC)

Solution:

The midpoint theorem states that the line joining the mid-points of two sides of a triangle is parallel to the third side and half of its length.

If one pair of the opposite side in a quadrilateral is parallel and equal to each other then it is a parallelogram. Diagonals separate the parallelogram into two triangles of equal areas.

Let's construct a diagram according to the given question.

i) In ΔABC, D and E are the mid-points of side BC and AC respectively.

Therefore, the line joining points D and E will be parallel to line AB and also half of it as per the midpoint theorem.

The mid-point of AB is Fand E is the mid-point of AC

∴ ED ║AB and ED = 1/2 AB

Since ED ║AB.

Therefore, ED ║FB and ED = FB [F is the midpoint of AB]

Since one pair of the opposite side in quadrilateral  BDEF is parallel and equal to each other, Therefore, BDEF is a parallelogram.

Similarly, we can prove, AEDF and CEFD are also parallelograms.