State and prove converse of BPT (basic proportionality theorem)


Statement of basic proportionality theorem (BPT)

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


Converse of Basic Proportionality Theorem

Suppose a line DE, intersects the two sides of a triangle AB and AC at D and E, such that;

AD/DB = AE/EC ……(1)

Assume DE is not parallel to BC. Now, draw a line DE’ parallel to BC.

Hence, by similar triangles,

AD/DB = AE’/E’C ……(2)

From eq. 1 and 2, we get;


Adding 1 on both the sides;

AE/EC + 1 = AE’/E’C +1

(AE +EC)/EC = (AE’+E’C)/E’C


So, EC = E’C

This is possible only when E and E’ coincides.

But, DE’//BC

Therefore, DE//BC.

Hence, proved.

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