`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`

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.

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 .

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 .

In DeltaPQR , a straight line parallel to QR intersects PQ and PR at the points D and E respectively. If PD=2.4cm , PE=3.2cm and ER=4. 8cm , then the length of PQ is

Debanjan drew a line segment PQ of which mid-point is R and two circles are drawn with PR adn PQ as diameter . Deberatee drew a straight line through the point which interects the first circle at the point S and the second circle at the point T. Prove that PS=ST

AD is a median of triangleABC . A straight line parallel to BC intersect AB and AC at P and Q respectively. Prove that AD bisects PQ.

Three points P,Q ,R lie on a circle. The two perpendiculars PQ and PR and the point P intersect the circle at the points S and T respectively. Prove that RQ=ST

Two two circle with their centre at P and Q intersect each other, at the point A and B . Through the point A , a straight line parallel to PQ intersects the two circles at the points C and D respectively . If PQ = 5 cm , then determine the lenght of CD .

Two circles intersect each other at the points G and H . A straight line is drawn through the point G which intersect two circles at the points P and Q and the straight line through the point H parallel to PQ intersects the two circles at the points R and S . Prove that PQ=RS .

Two circles intersect each other at the points P and Q. Two straight lines through P and Q intersect on circle at the points A and C and the other circle at B and D . Prove that AC||BD.