Sentence:  The Archimedean solids can all be inscribed in a sphere and in one of the Platonic solids. Their duals are the "Catalan solids", with faces that are congruent but not regular ("face-transitive"); indtead of circumspheres like the Archimedean solids, they have inspheres. The <WSD>midspheres</WSD>, touching the edges, are common to both of them. 
Meaning of the word marked with <WSD> tags:  (solid geometry) The sphere which, if it exists, is tangent to every edge of a given polyhedron and with respect to which the polyhedron's vertices are the inversion poles of the planes of the faces of its dual polyhedron (and vice versa).