What I think you do not understand is those curves are noisy, and you do not understand splines. The curves you drew are not noisy, no do they even go near the ends of the curve. In fact, they break away at some point, where you seem to think they should do so. A spline would never be able to know that however.
So you do not want to do cubic spline interpolation, because the extrapolation would be completely meaningless garbage where you try to extend the curves. You don't even want to do some sort of cubic spline fitting, because again, a spline would follow the curve, not peel away from it at some computationally unknowable spot.
Instead, take a short segment of the data near the break from each curve, in the part of the curve that you TRUST, not the part that you seem to want to ignore. Then just use a LOW order polynomial to fit. A straight line would be your best choice, given the noisiness of the curves. Then find the intersection of those straight lines.
