Question

State and prove Converse of Thales theorem​

Answer 1

Answer:

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Answer 2

Converse of Thales theorem

It states that if a line divides any two sides of the triangle in the same ratio, then the line is parallel to the third side

which means,

If AD/DB = AE/EC (Given)

=> DE is parallel to BC

Proof

Let us draw a triangle ABC and DE || BC

So, AD/DB = AE/EC {by thales theorem} ---> (1)

For some instance, let us assume that DE is not parallel to BC

Draw another point F on AC such that DF parallel to BC

By thales theorem,

AD/DB = AF/FC ----> (2)

From (1) and (2) :

AE/EC = AF/FC

Now adding 1 to both sides, we get

AE/EC + 1 = AF/FC + 1

=> (AE + EC) /EC = (AF + FC) /FC

=> AC/EC = AC/FC

* Cancelling AC from both side

=> EC = FC

This means that the length of EC is equal to FC

but we have drawn F not on E. Since the points doesn't coincide

∴ DE || BC