Finding Mid point between to two points on a sphere

I’m trying to make a trajectory for a part to go along between two points on a sphere. So I created a quadratic bezier curve to create that trajectory, however, I need to find the position of the middle point. I do this by lerping half-way between the start and end positions.

What I’m stuck on is how do I make this midpoint tangent to the sphere + the distance to be above the surface of the sphere.

I thought if I could make the point C face the center point, then the center point to face point C. I could make point C’s position relative to the center point and add the radius plus the distance above the surface of the sphere. That sorta of work but it wasn’t the results I was looking for, it also gave me another challenge to find the relative position between start and finish.

Assuming you have both positions of that you’re attempting to find the midpoint of, you can just lerp to the middle point.