`A B C D` is a quadrilateral and `B D` is one of its diagonals as shown in Figure. Show that `A B C D` is a parallelogram and find its area.

In Figure, A B C D is a parallelogram. Find the values of xa n dydot

If P is a point and A B C D is a quadrilateral and vec A P+ vec P B+ vec P D= vec P C , show that A B C D is a parallelogram.

If P is a point and A B C D is a quadrilateral and vec A P+ vec P B+ vec P D= vec P C , show that A B C D is a parallelogram.

In Figure, A B C D is a parallelogram and X ,Y are the mid=points of sides A B and D C respectively. Show that A X C Y is a parallelogram. Figure

In Figure, A B C D is a parallelogram and X\ a n d\ Y are points on the diagonal B D such that D X=B Y . Prove that A X C Y is a parallelogram (ii) A X=C Y ,\ A Y=C X A Y B\ ~= C X D (iv) A X D~=\ C Y B

If the diagonals A C ,B D of a quadrilateral A B C D , intersect at O , and seqarate the quadrilateral into four triangles of equal area, show that quadrilateral A B C D is a parallelogram. GIVEN : A quadrilateral A B C D such that its diagonals A C and B D intersect at O and separate it into four parts such that a r( A O B)=a r( B O C)=a r( C O D)=a r( A O D) TO PROVE : Quadrilateral A B C D is a parallelogram.

In Figure, A B C D is a cyclic quadrilateral. Find the value of a and b.

P is a point on the side B C of a triangle A B C such that A B=A P . Through A\ a n d\ C ,\ lines are drawn parallel to B C\ a n d\ P A , respectively, so as to intersect at D as shown in Figure. Show that A B C D is a cyclic quadrilateral.

If the diagonals A C ,\ B D of a quadrilateral A B C D , intersect at O , and separate the quadrilateral into four triangles of equal area, show that quadrilateral A B C D is a parallelogram.

In Figure, A B C D is a quadrilateral and B E || A C and also B E meets D C produced at E . Show that area of triangle A D E is equal to the area of the quadrilateral A B C D .