I made a program in Mathematica for calculating the point coordinates. The result is a large algebraic formula. I uploaded it to ideone
for you.
Here is the program, in case you have Mathematica at hand:
(*Load package*) Needs["VectorAnalysis`"] (*Define four lines, by specifying 2 points in each one*) Table[p[i, j] = {x[i, j], y[i, j], z[i, j]}, {i, 4}, {j, 2}]; (*Define the target point*) p0 = {x0, y0, z0}; (*Define a Norm function // using Std norm squared here*) norm[a_] := a[[1]]^2 + a[[2]]^2 + a[[3]]^2 (*Define a function for the distance from line i to point v used http://mathworld.wolfram.com/Point-LineDistance3-Dimensional.html (11) *) d[i_, v_] := norm[Cross[(v - p[i, 1]), (v - p[i, 2])]]/norm[p[i, 2] - p[i, 1]] (*Define a function for the sum of distances*) dt[p_] := Sum[d[i, p], {i, 4}] (*Now take the gradient, and Solve for Gradient == 0*) s = Solve[Grad[dt[p0], Cartesian[x0, y0, z0]] == 0, {x0, y0, z0}] (* Result tooooo long. Here you have it for downloading http://ideone.com/XwbJu *)
RESULT