Quickest way to find a point on a circumference Given the image below, A is the centre of the circle, B is a point on the circumference and AC and DB lie on parallel lines.
Knowing A, C, D and the radius of the circumference, I need to find the quickest way (in term of # of calculations) to find B. 

EDIT:
The position of C can vary, AC can also be parallel to x or y axis. This is for a program I'm making, I need to handle also corner cases in terms of calculations where the slope can be infinity (computers don't work well with infinity). In this context I found that using trigonometric functions I can achieve better results, but in this particular case I've come to no good solutions.

Let $A:={\bf 0}$, $C:={\bf c}$, $D:={\bf d}$, and $B:={\bf b}$. Then you want ${\bf b}={\bf d}+\lambda{\bf c}$ for some $\lambda>0$ and $|{\bf b}|^2=r^2$, where the radius $r>0$ is assumed to be given. Therefore
$$|{\bf d}|^2 + 2\lambda\langle{\bf d},{\bf c}\rangle +\lambda^2|{\bf c}|^2=r^2\ ,$$
which can be solved for $\lambda$ from the given data. Then ${\bf b}={\bf d}+\lambda{\bf c}$.
