Given two 2d circles that have intersected at two points, how do I find the distance along a ray that passes through the center of one of the circles so that when that circle is translated along that ray for that distance, the circles intersect exactly once?
In the above image, I'm trying to find the distance between the green points. The information I have is: both centers and radii, and the ray direction vector.
Update: Here is a diagram with the known/unknown lengths and angles:
In the triangle, the green line lengths are known and the green angle is known. The red ones are unknown. I thought about using the law of sines/cosines to solve for the angles, but it doesn't seem to me that I have enough information.