Basic Proportionality Theorem:

Basic Proportionality Theorem (can be abbreviated as BPT) states that, if a line is parallel to a side of a triangle which intersects the other sides into two distinct points, then the line divides those sides in proportion.

PROOF OF BPT

Given: In ΔABC, DE is parallel to BC

Line DE intersects sides AB and PQ in points D and E, such that we get triangles A-D-E and A-E-C.

To Prove: AD/BD=AE/CE

Construction: Join segments DC and BE

Proof:

In ΔADE and ΔBDE,

A(ΔADE)/A(ΔBDE)=AD/BD (triangles with equal heights)

In ΔADE and ΔCDE,

A(ΔADE)/A(ΔCDE)=AE/CE (triangles with equal heights)

Since ΔBDE and ΔCDE have a common base DE and have the same height we can say that,

A(ΔBDE)=A(ΔCDE)

Therefore,

A(ΔADE)/A(ΔBDE)=A(ΔADE)/A(ΔCDE)

Therefore,

AD/BD=AE/CE

Hence Proved.

The BPT also has a converse which states, if a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.

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