State and prove converse of BPT (basic proportionality theorem).
Converse 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.
Suppose a line , intersects the two sides of a triangle and at and , such that;
Assume is not parallel to . Now, draw a line parallel to .
Hence, by similar triangles,
From eq. and , we get;
Adding on both sides:
So,
This is possible only when and coincides.
But,
Therefore, .
Hence, proved.