Condition for External Tangency

Two circles touch externally iff the distance between centres equals the sum of radii. Exactly 3 common tangents in this case.

Let circle $C_1$ have centre $O_1$ and radius $r_1$, and circle $C_2$ have centre $O_2$ and radius $r_2$, with $d = O_1O_2$.

Two circles touch externally if they have exactly one common point and neither lies inside the other.

At the point of tangency $T$, the two radii $O_1T$ and $O_2T$ are collinear (both perpendicular to the common tangent at $T$) and point in opposite directions. Therefore $T$ lies on segment $O_1O_2$:

$$O_1T + O_2T = O_1O_2 \implies r_1 + r_2 = d$$

Common tangents in this configuration: 3 tangents exist — two external (direct) and one internal (transverse) passing through the point of tangency.

Converse: If $d = r_1 + r_2$, the two circles touch externally. The single point common to both is the point dividing $O_1O_2$ in the ratio $r_1 : r_2$ internally.