I have a circle with a known center and radius. I have an arc with a known radius and two points on its edge, where one end is at the same position as the circle's center. Is there a way to find their point of intersection that's simpler than turning this into two circles and intersecting those (and somehow figuring out which of the two points of intersection to use)?
In other words, the green bits of this diagram are known, and I'm trying to find the red:
From the given, I assume that J(h, k) can be found. Then, equation of OJ will be known. (This is especially simple if the y-ordinates of O and J are the same according to your drawing).
Applying the cosine law to the R-r-R triangle, $\theta$ can be found.
OX is the line that deviates $\theta$ from the line OJ. X is on the intersection of that line and the red circle.