Converse of Basic proportionality theorem or Thales theorem

 Imp rule use in this theorem is :-
:-You can add the same value to each side of an equation without changing the meaning of the equation:- 
For example:- Ax = By
 Ax+1=By+1

Theorem 6.2:-

 If a line divides any two sides of a triangle in the same ratio,then the line is parallel to the third side.

Proof:-

Given:-

△ABC and a line DE intersecting AB at D and AC at E,

Such that,    AD/DB = AE/EC 

To Prove:-   DE∥BC

Construction:- Draw DE'∥BC

 Proof:-

Since     DE'∥BC

(By the Theorem if a line is drawn parallel to one side of triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.)

AD/DB=AE'/E'C.........(1)

  and given that,
 AD/DB=AE/EC....(2)

From (1) and (2)
AE'/E'C=AE/EC

Adding 1 on both sides 
(AE'/E'C)+1=(AE/EC)+1
(AE'+E'C)/E'C=(AE+EC)/EC
 AC/E'C = AC/EC
1/E'C =1/EC
EC=E'C 
 Thus, E and E' coincide
 Since, DE'∥BC
∴ DE∥BC.
Hence, proved