ABCD is a parallelogram in which P is the midpoint of DC and Q is a point on AC such that CQ `=(1)/(4) AC`. If PQ produced meets BC at R, prove that R is the midpoint of BC.

In figure,P is the mid-point of BC, Q is the mid-point of AP, such that BQ produced meets AC at R .Prove that 3RA=CA

ABCD is a parallelogram.P is a point on AD such that AP=(1)/(3)AD and Q is a point on BC such that CQ=(1)/(3)BP. Prove that AQCP is a parallelogram.

In the figure ABCD is a trapezium in which side AB is parallel to side DC and E is the midpoint of side AD.If F is a point on the side BC such that the line segment EF is parallel to DC.Prove that Fis the mid-point of B BC and EF=(1)/(2)(AB+DC)

ABC is a triangle in which D is the midpoint of BC and E is the midpoint of AD. Prove that ar(triangleBED)=(1)/(4)ar(triangleABC) .

ABCD is a parallelogram. Any line through A cuts DC at a point P and BC produced at Q.Then,

squareABCD is a parallelogram. P is the midpoint of side CD. Seg BP meets diagonal AC at X. Prove that 3AX=2AC .

ABC is a triangle in which AB=AC and D is a point on AC such that BC^(2)=AC xx CD .Prove that BD=BC .

ABCD is a parallelogram. BC is produced to Q such that BC = CQ. Then

In the adjoining figure, ABCD is a parallelogram and E is the midpoint of AD. A line through D, drawn parallel to EB, meets AB produced at F and BC at L. Prove that (i) AF=2DC, (ii) DF=2DL