Inside of my game, I am in need of a way to test the collision between a line segment and a fast-moving circle. I would say I need a line v capsule collision however the problem is I need the collision point. None of the existing formulas I can find can do quite what I need. I have tried line x line intersection and just moving the lines by the radius of the capsule sort of work however then the collision point found around the edges is off. Additionally, I can't find a way to work back from the collision point to the point that the circle first collided at.
Above is sort of a picture of what I am hoping to achieve. The green capsule represents the circle's path.
So really the problem here is finding that collision point because a yes or no answer is trivial here. There are formulas for ray capsule or box capsule however none of them will get me this collision point.
Does anyone have any ideas on how I can figure this out?