A straight line drawn through `D` of the parallelogram `ABCD` intersects `AB` and the extended part of `CB` at the points `E` and `F` respectively. Prove that `AD:AE=CF:CD`.

In trapezium ABCD , AB and DC are parallel. A straight line parallel to AB intersects AD and BC at the points E and F respectively. Prove that DE:EA=CF:FB .

DeltaABC and DeltaDBC are situated on the same base BC and on the same side of BC . E is any point on the side BC . Two line through the point E and parallel to AB and BD intersects the sides AC and DC at the points F and G respectively. Prove that AD||FG .

Ankita drew two circles which intersect each other at the points P and Q. Through the point P two straight lines are draw so that they intersect one of the circle at the points. A and B and the ohte circle at the points C and D respectively. Prove that angle AQC= angle BQD .

The diagonals AC and BD of the parallelogram ABCD intersect each other at O . Any straight line passing through O intesects the sides AB and CD at the points P and Q respectively . Prove that OP = OQ.

Two straight lines are drawn through any point X and exterior to a circle to intersect the circle at points A, B and points C, D respectively . Prove that DeltaXACandDeltaXBD are equiangular.

The straight line parallel to the side BC of DeltaABC intersects the sides AB and AC at the points D and E respectively. Then (AB)/(BD)=(AC)/(CE) .

Two parallel straight lines intersect three concurrent straight lines at the points A , B , C and X,Y,Z respectively. Prove that AB : BC=XY : YZ .

ABCD is parallelogram A circle passing through the points A and B intersect the sides AD and BC at the points E and F repectively . Prove that the four points E, F , C ,D are concyclic.

PQRS is a parallelogram. If a straight line EF through S intersects PQ and extended RQ at the points X and Y respectively, then prove that PS:PX=QY:QX=RY:RS

Two tangents AB and AC drawn from an external point A of a circle touch the circle at the point B and C. A tangent drawn to a point X lies on minor arc BC intersects AB and AC at the points D and E respectively. Prove that peremeter of triangleADE = 2AB .