`A B C D` is a quadrilateral; `P , Q , Ra n dS` are the points of trisection of side `A B ,B C ,C Da n dD A` respectively and are adjacent to `Aa n dC` ; prove that `P Q R S` is parallelogram.

ABCD is a quadrilateral; P,Q,R and S are the points of trisection of side AB,BC,CD and DA respectively and are adjacent to A and C; prove that PQRS is parallelogram.

A B C D is a rhombus and P ,\ Q ,\ R ,\ S are the mid-points of A B ,\ B C ,\ C D ,\ D A respectively. Prove that P Q R S is a rectangle.

A B C D is a cyclic quadrilateral. A Ba n dD C are produced to meet in Edot Prove that E B C-E D Adot

Prove that the figure formed by joining the mid-points of the pair of consecutive sides of a quadrilateral is a parallelogram. OR B A C D is a parallelogram in which P ,Q ,R and S are mid-points of the sides A B ,B C ,C D and D A respectively. A C is a diagonal. Show that : P Q|| A C and P Q=1/2A C S R || A C and S R=1/2A C P Q=S R (iv) P Q R S is a parallelogram.

P\ a n d\ Q are the points of trisection of the diagonal B D of a parallelogram A B C Ddot Prove that C Q is parallel to A Pdot Prove also that A C bisects P Q

In a A B C ,\ E\ a n d\ F are the mid-points of A C\ a n d\ A B respectively. The altitude A P\ to\ B C intersects F E\ at Qdot Prove that A Q=Q P

In Delta P Q R , if P Q=Q R and L ,M and N are the mid-points of the sides P Q ,Q R and R P respectively. Prove that L N=M Ndot

In Figure, A B C D is a parallelogram. E\ a n d\ F are the mid-points of the sides A B\ a n d\ C D respectively. Prove that the line segments A F\ a n d\ C E trisect (divide into three equal parts) the diagonal B D .

If D ,E ,F are the mid-points of the sides B C ,C Aa n dA B respectively of a triangle A B C , prove by vector method that A r e aof D E F=1/4(a r e aof A B C)dot

A B C D is a parallelogram. L\ a n d\ M are points on A B\ a n d\ D C respectively and A L=C M . Prove that L M\ a n d\ B D bisect each other.