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.

In DeltaABC a straight line parallel to BC intersects the sides AB and AC at the points P and Q respectively. If AP=QC , AB=12cm , AQ=2cm , then find the value of CQ .

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 circles have been drawn with centres A and B which touch each other externally at the point O. A straight line is drawn passing thrugh the point O and intersects the two circles at P and Q respectively. Prove that AP ||BQ.

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.

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.

The diagonals AC and BD of a quadrilateral ABCD intersects each other at O such that DeltaAOD=DeltaBOC . Prove that ABCD is trapezium.

In two circles , one circle passes through the centre O of the other circle and they intersect each other at the points A and B . A straight line passing through A intersect the circle passing through O at the point P and the circle with centre at O at the point R . BY joining P , B and R , B prove that PR= PB.

Two circles intersect each other at the points P and Q. A straight line passing through P intersects one circle at A the other circle at .A straight line passing through Q intersect the first circle at C and the second circle at D. Prove that AC||BD .

In triangleABC , a straight line parallel to BC intersects AB and AC at P and Q respectively. If AB = 3PB, then PQ : BC =

Two circle intersect each other at A and B and a straight line parallel to AB intersects the circles at C,D,E,F. Prove that CD = EF.