Problem: Given n lines, determine a point minimizing the summation of the

distances from the point to the n lines.

This is a Least Absolute Deviations problem, which are normally solved by simplex methods; see the Wikipedia page for more details. If an O(n^3) running time is acceptable to you, however, a minimal solution will always be found at the intersection of two of the lines, so you can just check all the intersection points if you like.

Ben