`D` is a point on the circumcircle of ` A B C` in which `A B=A C` such that `B\ a n d\ D` are on the opposite side of line `A C`. If `C D` is produced to a point `E` such that `C E=B D ,` prove that `A D=A E`.

A B C D is a parallelogram. If A B is produced to E such that B E=A B . Prove that E D bisects B C .

A B C D is a parallelogram. A B is produced to E so that B E=A B . Prove that E D bisects B C .

A D\ a n d\ B E are respectively altitudes of A B C such that A E=B D . Prove that A D=B E .

In Figure, A D=A E\ a n d\ D\ a n d\ E are points on B C such that B D=E Cdot Prove that A B=A C

D and E are points on equal sides A B and A C of an isosceles triangle A B C such that A D=A E . Prove that B ,C ,D ,E are concylic.

In Figure, A C > A B\ a n d\ D is the point on A C such that A B=A Ddot Prove that B C > C D

If D and E are points on sides A B and A C respectively of a triangle A B C such that D E || B C and B D=C E . Prove that A B C is isosceles.

In Figure, the side B A\ a n d\ C A have been produced such that B A=A D\ a n d\ C A=A Edot Prove that segment D E||B C

D\ a n d\ E are, respectively, the points on equal sides A B\ a n d\ A C of an isosceles triangle A B C such that B ,\ C ,\ E ,\ a n d\ D are concyclic as shown in Figure. If O is the point of intersection of C D\ a n d\ B E , prove that A O is the bisector of line segment D E

A B C is a triangle. D is a point on A B such that AD=1/4A B\ a n d\ E is a point on A C such that A E=1/4A C . Prove that D E=1/4B C