The midscribability theorem, which was first proved by O. Schramm, states that: given a smooth strictly convex body $$K\subset {\mathbb {R}}^{3}$$ K ⊂ R 3 and a convex polyhedron $$P$$ P , there exists a convex polyhedron $$Q\subset {\mathbb {R}}^3$$ Q ⊂ R 3 combinatorially equivalent to $$P$$ P which midscribes $$K$$ K . Here the word “midscribe” means that all its edges are tangent to the boundary surface of $$K$$ K . By using the intersection number technique, together with the Teichmüller theory of packings, this paper provides an alternative approach to this theorem. Furthermore, by combining Schramm’s method with the above ones, we obtain a rigidity result as well. That is, such a polyhedron is unique under the normalization condition.