Question

abc is a triangle the bisector of exterior angle b and the bisector of c intersect each other at d . prove angle bdv =1/2angle a

Answer 1

Let D, E and F be feet of perpendiculars from O onto ray AB, BC and ray AC respectively.
Since OB bisects angle DBE, OD=OE
Since OC bisects angle ECF, OE=OF
Therefore OD=OF, and it is easy to prove that triangles ODA and OFA are congruent. And therefore OA bisects angle A.
The circle with centre O and radius OA will touch the sides (or extended sides) at D, E and F, much like an incircle. This circle is called an excircle, since it lies outside the triangle. In fact, for any given triangle, it is possible to draw three excircles, one on each side.