`A B C D` is a parallelogram whose diagonals `A C` and `B D` intersect at `Odot` A Line through `O` intersects `A B` at `P` and `D C` at `Qdot` Prove that `a r( P O A)=a r( Q O C)dot`

A B C D is a parallelogram whose diagonals A C\ a n d\ B D intersect at O . A line through O intersects A B\ a t\ P\ a n d\ D C\ at Q . Prove that a r\ (\ /_\P O A)=\ a r\ (\ /_\Q O C) .

The diagonals of a parallelogram A B C D intersect at OdotA line through O intersects A B at X and D C at Ydot Prove that O X=O Ydot

The diagonals of a parallelogram A B C D intersect at O. A line through O intersects A B\ at X\ a n d\ D C at Y . Prove that O X=O Y .

The diagonals of a parallelogram A B C D intersect at Odot A line through O meets A B in X and C D in Ydot Show that a r( A X Y D)=1/2(A R ^(gm)A B C D)

The diagonals of a parallelogram A B C D intersect at Odot A line through O meets A B\ in x\ a n d\ C D in Ydot Show that a r\ (A X Y X)=1/2(a r|""|^(gm)\ \ A B C D)

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.

A B C D is a trapezium with A B||D C . A line parallel to A C intersects A B\ at X\ a n d\ B C at Ydot Prove that a r( A D X)=a r(\ A C Y)dot

In a parallelogram A B C D diagonals A C and B D intersect at O and A C=6. 8c m and B D=13. 6c m . Find the measures of O C and O D .

A B C D is a parallelogram whose diagonals intersect at O . If P is any point on B O , prove that: a r\ (triangle \ A B P)=\ a r\ (triangle \ C B P)

Let A B C D be a p[arallelogram whose diagonals intersect at P and let O be the origin. Then prove that vec O A+ vec O B+ vec O C+ vec O D=4 vec O Pdot