`Da n dE` are, respectively, the points on equal sides `A Ba n dA C` of an isosceles triangle `A B C` such that `B , C , Ea n dD` are concyclic as shown in Figure. If `O` is the point of intersectionof `C Da n dB E ,` prove that `A O` is the bisector of line segment `D E` .

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

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.

Da n dE are respectively the points on the side A Ba n dA C of a A B C such that A B=5. 6 c m ,A D=1. 4 c m ,A C=7. 2 c m and A E=1. 8 c m , show that D E ||B Cdot

If D , E ,a n dF are three points on the sides B C ,A C ,a n dA B of a triangle A B C such that A D ,B E ,a n dC F are concurrent, then show that BD*CE*AF=DC*EA*FB .

In Figure, A D\ a n d\ B E are respectively altitudes of an isosceles triangle A B C with A C=B Cdot Prove that A E=B D

In an isosceles triangle A B C with A B=A C , a circle passing through B and C intersects the sides A Ba n dA C at Da n dE respectively. Prove that DE∣∣BC Bdot

In Figure, A B=C D . Prove that B E=D E\ a n d\ A E=C E , where E is the point of intersecting of A D\ a n d\ B C .

The bisector of /_B of an isosceles triangle A B C with A B=A C meets the circumcircle of A B C at P as shown in Figure. If A P\ a n d\ B C produced meet at Q , prove that C Q=C A .

In an isosceles triangle A B C with A B=A C , a circle passing through B\ a n d\ C intersects the sides A B\ a n d\ A C at D\ a n d\ E respectively. Prove that D E || B C .

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.