Find circle for two points, one with given angle.

Question

I have point A and B. I also have a vector v. How can I mathematically find a circle whose tangent at point C has the same angle as v where point C is the same as B and the circle also contains point A.

Here is an illustration:

Black dot is A Red dot is B Orange arrow is vector v Green circle is on both A and B, while one of it's tangents is on v. Black line is the tangent.

Answer 1

The center is equidistant from $A$ and $B$, so it lies on the perpendicular bisector of these two points. Call the bisector $\ell_1$

Any radius is perpendicular to a tangent vector. So, the center lies on a line through $B$ perpendicular to the tangent. Call this second line $\ell_2$.

The center of the circle must be at the (generally uniquely existent) intersection of lines $\ell_1$ and $\ell_2$. The radius of the circle is simply the distance from this point (the center) to either $A$ or $B$.