Proving a graph is planar - mutually tangent circles in a plane 
Let $C_1,\dots ,C_n$ be circles in the plane with pairwise disjoint
  interiors. Define the tangency graph to have $n$ vertices such
  that vertices are adjacent if the corresponding circles are tangent to
  each other. Prove this graph is planar.

It looks like I have to prove there can't be minor $K_5$'s or $K_{3,3}$'s. For $K_5$, I thought to prove there's no $K_4$ anywhere. Thing is, I don't even know how to rigorously prove we can't have four pairwise tangent circles in the plane. This doesn't even cover minors and I have no idea how to tackle $K_{3,3}$'s... How to solve this problem? How should such problems be approached?

It's obvious: Take the centers $v_i$ $(1\leq i\leq n)$ of the circles as vertices and connect two vertices by a straight segment if the corresponding circles are tangent to each other.
Or have I missed something?
