How to find center of an arc given a geographical start point, end point and arc angle?

Question

I am working in a way of drawing transitions in some kind of "Flight Plan" between two "waypoints" using the heading at the beginning of the arc and at the end of an arc. Given an arbitrary constant radius arc, where I know the following values:

1. starting WP Coordinates($ \phi_{0}, \lambda_{0} $)
2. end WP Coordinates($ \phi_{1}, \lambda_{1} $)
3. arc direction (i. e. clockwise or counterclockwise).
4. Because of the headings at the beginning and at the end, I know the angle of the arc will be $\theta = Heading_{2}-Heading_{1}$

I know graphically, that because that angle is also the arc angle, I can make an arc "fit" where two lines opening that angle intersect the two waypoints. I don't know how to calculate it given their coordinates though. I think I could assume a flat earth for small distances.

I would like to know how to approach the calculation of the arc center (as coordinates) and with this the radius of the arc, or of course, the other way around! Thanks!