Number of Common Tangents

Number of common tangents as a function of the distance d between centres relative to radii r₁ and r₂.

For two circles with radii $r_1, r_2$ and distance $d$ between centres:

External tangents (direct common tangents): Lines that do not pass between the two circles. They exist whenever the circles do not overlap. Their intersection point (external centre of similitude) divides $O_1O_2$ externally in the ratio $r_1 : r_2$.

Internal tangents (transverse common tangents): Lines passing between the circles. They exist only when the circles are external to each other (including external tangency). Their intersection (internal centre of similitude) divides $O_1O_2$ internally in the ratio $r_1 : r_2$.

Length of an external common tangent:

$$\ell_{\text{ext}} = \sqrt{d^2 - (r_1 - r_2)^2}$$

Length of an internal common tangent:

$$\ell_{\text{int}} = \sqrt{d^2 - (r_1 + r_2)^2}$$

(defined only when $d > r_1 + r_2$)

Derivation of external tangent length: Drop perpendiculars from $O_1$ and $O_2$ to the tangent. They are parallel and equal to $r_1$ and $r_2$ respectively. The gap between the feet equals $\ell_{\text{ext}}$. Construct a rectangle to reduce to a right triangle with hypotenuse $d$ and one leg $|r_1 - r_2|$. By Pythagoras: $\ell_{\text{ext}}^2 = d^2 - (r_1 - r_2)^2$.