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

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 .

E and F are the mid-points of the sides AB and CD of a parallelogram ABCD. Prove that the line segment AF and CE trisects BD in three equal parts.

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 a parallelogram A B C D ,E ,F are any two points on the sides A B and B C respectively. Show that a r( A D F)=a r( D C E)dot

A B C D is a parallelogram, E\ a n d\ F are the mid-points of A B\ a n d\ C D respectively. G H is any line intersecting A D ,\ E F\ a n d\ B C at G ,\ P\ a n d\ H respectively. Prove that G P=P H

A B C D is a parallelogram and E\ a n d\ F are the centroids of triangles A B D\ a n d\ B C D respectively, then E F= (a) A E (b) B E (c) C E (d) D E

D ,\ E ,\ F are the mid-points of the sides B C ,\ C A and A B respectively of a A B C . Determine the ratio of the areas of D E F and A B C .

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

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

A B C D is a parallelogram X and Y are the mid-points of B C and C D respectively. Prove that a r( A X Y)=3/8a r(^(gm)A B C D) GIVEN : A parallelogram A B C D in which X and Y are the mid-points of B C and C D respectively. TO PROVE : a r( A X Y)=3/8a r(^(gm)a b c d) CONSTRUCTION : Join B Ddot