Two lines `A B\ a n d\ C D` intersect at `O` such that `B C` is equal and parallel to `A Ddot` Prove that the lines `A B\ a n d\ C D` bisect at `O`

Two lines A B and C D intersect at O such that B C is equal and parallel to A Ddot Prove that the lines A B and C D bisect at Odot

Suppose line segments A B\ a n d\ C D intersect at O in such a way that A O=O D\ a n d\ O B=O C . Prove that A C=B D but A C may not be parallel to B D .

Two segments A C and B D bisect each other at O . Prove that A B C D is a parallelogram.

Two segments A C and B D bisect each other at O . Prove that A B C D is a parallelogram.

Suppose line segments A B and C D intersect at O in such a way that A O=O D and O B=O C . Prove that A C=B D but A C may not be parallel to B D .

Two circles whose centres are O and O ' intersect at Pdot Through P , a line l parallel to O O ' intersecting the circles at C\ a n d\ D is drawn. Prove that C D=2\ O O '

Two circles whose centres are O and O ' intersect at Pdot Through P , a line l parallel to O O ' intersecting the circles at C\ a n d\ D is drawn. Prove that C D=2\ O O '

Diagonals A C\ a n d\ B D of a quadrilateral A B C D intersect at O in such a way that a r\ ( A O D)=a r\ (\ B O C)dot Prove that A B C D is a trapezium.

Diagonals A C and B D of a quadrilateral A B C D intersect at O in such a way that a r(triangle A O D)=a r(triangle B O C) . Prove that A B C D is a trapezium.

A B C D is a parallelogram whose A C\ a n d\ B D intersect at Odot A line through O intersects A B\ a t\ P\ a n d\ D C\ at Qdot Prove that a r\ (\ P O A)=\ a r\ (\ Q O C)dot