# calculate the coordinates of the intersection between a bisector and a sector

I have a sector and would like a formula that gives the intersection between the bisector and the arc. here's a graph of the situation:

the point B is the center of a circle of radius AB and the BD ray is the bisector of the ABC angle, knowing the coordinates of A, B, C as well as the ABC angle and the AB length I would like to know how I could calculate the coordinate of the D point. thanks by advance for your help.

• It would help if you told us what did you try to solve this problem. – Moishe Kohan Aug 25 '14 at 20:49
• I try to put labels at the exterior of the circle for each slices of a pie chart. the coordinate of D would be calculated for each slices and be used for putting the corresponding label there. – Graou13 Aug 25 '14 at 20:54

Find the midpoint $E$ of $AC$, then take $\lambda$ as $\frac{AB}{BE}$.
You will have $$D = B + \lambda(E-B).$$