In the picture below, the diagonals of the trapezium `ABCD` intersect at P.
Prove that
`PAtimesPD=PBtimesPC`.
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In the picture below, the side AC of the triangle ABC is extended by D, by adding the length of the side CB. Then the line through C parallel to DB is drawn to meet AB at E. Prove that CE bisects angle C

In the figure the diagonals AC and BD intersect at P. The area of PAB is 30 cm^2 , area of PDA is 60 cm^2 and area of PBC is 40 cm^2 . What is PD:PB ?

In the figure the diagonals AC and BD intersect at P. The area of PAB is 30 cm^2 , area of PDA is 60 cm^2 and area of PBC is 40 cm^2 . What is the area of PDC

In the figure the diagonals AC and BD intersect at P. The area of PAB is 30 cm^2 , area of PDA is 60 cm^2 and area of PBC is 40 cm^2 . What is PA:PC ?

Prove that diagonals of trapezium cut each other proportionally.

In the picture below, two vertices of a parallelogram are joined to the mid points of two sides. Prove that these lines divides the diagonal in the picture into three equal parts. .

In the two pictures below, the squares are of the same size. Prove that the green regions are of the same area.

In the picture below, two lines are drawn from the top vertex of a triangle to the bottom side: Prove that the ratio in which these lines divide the length of the bottom side is equal to the ratio of the area of the three smaller triangles in the picture.

The picture on the right shows two circles centred on the same point and a line intersecting them.Prove that the parts of the line between the circles on either side are equal.

In the figure, A B C D is a trapezium. Diagonal A C and B D intersect at M. Area of triangle ABC is 24 cm^2 and area of triangle AMB is 10cm^2. a) Find the area of triangle BMC. b) How much is the area of triangle AMD? Write reason.