That's a very interesting problem. I have a couple of questions for you:
1. Does the "point of best fit" have to be on at least one of the lines? On at least two of the lines? Or could it be anywhere?
2. How are you defining "closest"? Since you only have two variables, x and y, in your equations, are you in just the xy plane? If so, are you measuring distance using Euclidean distance?