The drawing below shows that the multiples are ratios of areas, and
that the interesection point is:
c = u (q^v)/(u^v) + v (p^u)/(v^u)
You can define the lines by dragging p and u, or q and v. The bivector moments (see the line demo) define the desired ratios, as their reshaping parallel to u and v is meant to show. Make sure you also make configurations with opposite directions of u and v, to convince yourself of the universality of this construction.
What if the lines are not in a plane? Then you can
compute their separation vector.
Created with Cinderella by Leo Dorst 20010719.