Is there a formula for finding the centers of the faces of a platonic solid? Is there a formula for finding the centers of the faces of a platonic solid given the center of the first (origin) face to be $P_0(x_0,y_0,z_0)$?

you can find the formula(coxeter) for the radius of the inner sphere of regular solids. the contact point of the plane of the face and the inner sphere is the center of that face.