The medians BE and CF of the `DeltaABC` intersect each other at G and the line segment EF intersects AG at O. Prove that OA = 3.OG.

The two medians BE and CF of DeltaABC intersect each other at the point G and if the line segment FE intersects the line segment AG at the point O , then prove that AO=3OG .

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 medians BE and CF of Delta ABC intersect at G. Prove that area of the quadrilateral AFGE=(1)/(3)Delta ABC.

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

Two chords AB and CD of the circle with centre at O intersect each other at an internal point P of the centre. Prove that angle AOD+ angle BOC=2 angle BPC

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.

The diagonals of the square ABCD intersect each other at O .Let OPbotAB is constructed. Prove that angleAOP =anglePBO

D and E are the mid-points of AB and AC respectively of the DeltaABC. AP is the median of BC. AP intersects DE at Q. Prove that DQ = QE and AQ = QP.

The diagonals of the quadrilateral ABCD intersect each other at O in such a way that (AO)/(BO)=(CO)/(DO) . Prove that ABCD is a trapezium.