State "Basic proportionality theorem"

State and prove Basic proportionality theorem

Study the following figure. Write the ratios in relation to basic proportionality theorem and its corollarise, in terms of a, b, c and d.

State and prove Bionomial theorem for any positive integer n.

State and prove pythagoras theorem .

If the diagonals of a quadrilateral divide each other proportionally, then prove that the quadrilateral is a trapezium.

If a straight line divdes two sides of a triangle proportionally, then the straight line a parallel to third sides, (Converse of Thales theorem). Prove.