In ` A B C ,D` is the mid-point of `B Ca n dE D` is the bisector of the `/_A D Ba n dE F` is drawn parallel to `B C` cutting `A C` in `F` . Prove that `/_E D F` is a right angle.

In o+ABC,D is the mid-point of BC and ED is the bisector of the /_ADB and EF is drawn parallel to BC cutting AC in F. Prove that /_EDF is a right angle.

The sides A B\ a n d\ C D of a parallelogram A B C D are bisected at E\ a n d\ Fdot Prove that E B F D is a parallelogram.

Let A B C be an isosceles triangle in which A B=A Cdot If D ,\ E ,\ F be the mid-points of the sides B C ,\ C A\ a n d\ A B respectively, show that the segemtn A D\ a n d\ E F bisect each other at right angles.

In Fig.87, the corresponding arms of /_A B C\ a n d\ /_D E F are parallel. If /_A B C=75^0, find the /_D E Fdot

A B C D is a rhombus, E A B F is a straight line such that E A=A B=B F . Prove that E D and F C when produced meet at right angles.

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

A B C D is a parallelogram. E is a point on B A such that B E=2\ E A\ a n d\ F is a point on D C such that D F=2\ F Cdot Prove that A E\ C F is a parallelogram whose area is one third of the area of parallelogram A B\ C Ddot

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 square. E is the mid-point of B C and F is the mid-point of C D . The ratio of the area of triangle A E F to the area of the square A B C D is (a) 1 : 2 (b) 1 : 3 (c) 1 : 4 (d) 3 : 8

A B C D is a parallelogram, G is the point on A B such that A G=2\ G B ,\ E\ is a point of D C such that C E=2D E\ a n d\ F is the point of B C such that B F=2F Cdot Prove that: a r\ (\ E B G)=a r\ (\ E F C)