B.P.T.

Basic proportionality theoram (B.P.T.)
Statement:

If the line parallel to the side of the triangle intersects other sides in two distinct points then the other sides are divide in the same ratio by it.

     

Given:

In ∆PQR, line l॥ side QR and intersects side PQ and side PR in points M and N respectively. 

     P-M-Q and P-N-R.

To Prove:
               
Construction:

Join seg QN and seg RM

 Proof:

In ∆PMN and ∆QMN,

ஃ.......(1) (triangles having equal heights).

      In ∆PMN and ∆RMN,
ஃ........(2) (triangles having equal heights).
 
ஃ A(∆QMN)=A(∆RMN)....(3) (triangles with common base and equal height).

ஃ......(4). (from (1)..(2).&(3).)  
ஃ (from (1)..(2)..&..(4)...)

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Thank you
Ashok Kadam.