Number of common tangents based on distance d between centres and radii r1, r2: d > r1+r2: 4 tangents (2 direct + 2 transverse). d = r1+r2: 3 tangents. |r1-r2| < d < r1+r2: 2 tangents (direct only). d = |r1-r2|: 1 tangent. d < |r1-r2|: 0 tangents. Direct common tangents divide the line segment joining centres externally in ratio r1:r2. Transverse common tangents divide it internally in ratio r1:r2.