`A B C D` is a parallelogram. `P` is a point on `A D` such that `A P=1/3\ A D\ a n d\ Q` is a point on `B C` such that `C Q=1/3B C`. Prove that `A Q C P` is a parallelogram.

A B C D is a parallelogram. P is a point on A D such that A P=1/3A D and Q is a point on B C such that C Q=1/3B P . Prove that A Q C P is a parallelogram.

A B C D is a parallelogram. E is a point on B A such that B E=2\ E A\ a n d\ F is a point on D C such that D F=2\ F C . Prove that A E C F is a parallelogram whose area is one third of the area of parallelogram A B C D .

In Figure, A B C D is a parallelogram in which P is the mid-point of D C\ a n d\ Q is a point on A C such that C Q=1/4A C . If P Q produced meets B C\ a t\ R , prove that R is a mid-point of B C .

A B C D is a parallelogram. P is the mid-point of A BdotB D\ a n d\ C P intersect at Q such that C Q : Q P=3: 1. If a r\ ( P B Q)=10\ c m^2, find the area of parallelogram A B C D

In Figure, A B C D is a parallelogram in which P is the mid-point of D C and Q is a point on A C such that C Q=1/4A Cdot If P Q produced meets B C at Rdot Prove that R is a mid-point of B Cdot

In Figure, X ,\ Y are the mid-points of opposite sides A B\ a n d\ D C of a parallelogram A B C D. A Y\ a n d\ D X are joined intersecting in P ; C X\ a n d\ B Y are joined intersecting in Q . Show that (i) A X C Y is a parallelogram (ii) D X B Y\ is a parallelogram (iii) P X Q Y is a parallelogram

The diagonals of a parallelogram A B C D intersect at a point O . Through O , a line is drawn to intersect A D\ at P and B C at Q . Show that P Q divides the parallelogram into two parts of equal area.

The diagonals of a parallelogram A B C D intersect at a point Odot Through O , a time is drawn to intersect A D at P and B C at Q . Show that P Q divides the parallelogram into two parts of equal area.

In Fig. 9.32, ABCD is a parallelogram and BC is produced to a point Q such that A D\ =\ C Q . If AQ intersect DC at P, show that a r\ (B P C)\ =\ a r\ (D P Q)dot

A B C is a triangle in which A B=A C\ a n d\ P is a point on A C . Through C a line is drawn to intersect B P produced at Q such that /_A B Q=\ /_A C Q . Prove that: /_A Q C=90^@+1/2/_B A C