What you have is a line intersecting a circle (or sphere in 3 dimensions). A point of intersection (x,y) is both on the line and on the circle, so satisfies the equation for both. You end up with a non-linear system of 2 equations (one quadratic, one linear) in two unknowns (x and y). Solving that system gives you your points of intersection:
Line AC: line through (100, 100) and (461, 53):
slope = , so it's , so
System of equations is:
Plug the expression for y given by the 2nd equation into the first equation, and, after expanding and then collecting like terms, it becomes a quadratic equation for x, which has two solutions (hopefully - according to the picture). For each solution for x, use the second equation to find the value of y. This is how you find the two points of intersection (x,y).