Any point `X` inside ` D E F` is joined to its vertices. From a point `P` in `D X ,\ P Q` is drawn parallel to `D E` meeting `X E` at `Q` and `Q R` is drawn parallel to `E F` meeting `X F` in `R` . Prove that `P R D F` .

In the figure, a point O inside triangleABC is joined to its vertices. From a point D on AO, DE is drawn parallel to AB and from a point E on BO, EF is drawn parallel to BC. Prove that DF || AC.

P is any point on base B C of A B C\ a n d\ D is the mid-point of B CdotD E is drawn parallel to P A to meet A C at Edot If a r\ ( A B C)=12 c m^2 , then find area of E Q C

Two triangle ABC and D B C lie on the same side of the base B C . From a point P on B C ,P Q A B AND P R B D ARE DRAWN. They meet A C in Q and D C in R respectively. Prove that Q R A Ddot

P is the mid-point of side A B of a parallelogram A B C D . A line through B parallel to P D meets D C at Q\ a n d\ A D produced at Rdot Prove that: A R=2B C (ii) B R=2\ B Q

In Figure A B C D\ a n d\ A E F D are two parallelograms. Prove that: P E=F Q

P Q R S is a parallelogram. P X\ a n d\ Q Y are respectively, the perpendiculars from P\ a n d\ Q to S R\ a n d\ R S produced. Prove that P X=Q Y

The side A B of a parallelogram A B C D is produced to any point Pdot A line through A parallel to C P meets C B produced in Q and the parallelogram P B Q R completed. Show that a r(^(gm)A B C D)=a r(^(gm)B P R Q)dot CONSTRUCTION : Join A C and PQ. TO PROVE : a r(^(gm)A B C D)=a r(^(gm)B P R Q)

If A B C\ a n d\ D E F are two triangles such that A B ,\ B C are respectively equal and parallel to D E ,\ E F , then show that Quadrilateral A B E D is a parallelogram Quadrilateral B C F E is a parallelogram A C=D F (iv) A B C\ ~=\ D E F

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