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 /_\A B C ,D is the mid-point of BC and ED is the bisector of the /_ADB and EF is drawn parallel to B C cutting A C in F . Prove that /_E D F 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\ F . Prove that E B F D is a parallelogram.

In Figure, A B C D isa trapezium in which side A B is a parallel to side D C and E is the mid-point of side A D . If F is a point on the side B C such that the segment E F is parallel to side D C . Prove that F is the mid point of B C and E F=1/2(A B+D C) .

A D ,\ B E\ a n d\ C F , the altitudes of A B C are equal. Prove that A B C is an equilateral triangle.

A D ,\ B E\ a n d\ C F , the altitudes of A B C are equal. Prove that A B C is an equilateral triangle.

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 Q . Prove that A Q=Q P .

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

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 C . Prove that A E C F is a parallelogram whose area is one third of the area of parallelogram A B C D .

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 a \ A B C ,\ A D bisects /_A\ a n d\ /_C >/_Bdot Prove that /_A D B >\ /_A D C .