The relative position of two circles depends on d (distance between centres) compared to r1+r2 and |r1-r2|. External non-intersection: d>r1+r2 (4 common tangents). External tangency: d=r1+r2 (3 tangents). Intersection: |r1-r2|<d<r1+r2 (2 tangents). Internal tangency: d=|r1-r2| (1 tangent). One inside other: d<|r1-r2| (0 tangents). Concentric: d=0. The common chord of intersecting circles is S1-S2=0. For tangent circles, this becomes the common tangent at the point of contact. The length of direct common tangent is sqrt($d^{2-}$(r1-r2)^2) and transverse common tangent is sqrt($d^{2-}$(r1+r2)^2). Direct tangents pass through the external centre of similitude (divides C1C2 externally in r1:r2). Transverse tangents pass through the internal centre of similitude (internal division).