Let $ O$ be the center of the sphere to which the edges are tangent, and let $ R$ be its radius. Let $ D$ be the side length of an isosceles triangle whose base is the length of an edge of $ P$ and whose height is $ R$. for each vertex $ A$ of $ P$, color it black, white or red according to wheter $ AO$ is less than, greater than or equal to $ D$. Prove that a neighbor of a red vertex must be red, a neighbor of a black vertex is white, and a vertex of a white vertex is black. This proves that if there's one red vertex, they must all be red, so the sphere with center $ O$ and radius $ D$ passes through all the vertices. If there is no red vertex, they are all black/white, which is impossible since there's an odd cycle.