Two circles, each of radius 4 cm, touch externally. Each of these two circles is touched externally by a third circle. If these three circles have a common tangent, then the radius of the third circle, in cm, is

Let the two equal circles have centers $C_1$ and $C_2$ with radii $R_1=R_2=R=4$ cm. Let the third circle have center $C_3$ and radius $r$.

Since all three circles have a common tangent line, the third smaller circle lies in the gap between the two larger circles and the tangent line. Let the common tangent line be horizontal.

The distance between the contact points of the two larger circles on the tangent line is equal to the distance between their centers, which is $2R=8$ cm.

For any two externally touching circles of radii $A$ and $B$ that also touch a common tangent line, the distance $d$ between their points of contact on the line is given by:
$d=2\sqrt{AB}$

Applying this relation to the third circle of radius $r$ touching each of the larger circles of radius $R$, the distance between the contact points of the two larger circles is the sum of the contact distances from the smaller circle to each of the larger ones:
$2\sqrt{R\cdot R}=2\sqrt{Rr}+2\sqrt{Rr}$

Simplifying the equation:
$2R=4\sqrt{Rr}$
$R=2\sqrt{Rr}$
$\sqrt{R}=2\sqrt{r}$

Squaring both sides:
$R=4r$

Given $R=4$ cm:
$4=4r\Rightarrow r=1$ cm.

Thus, the radius of the third circle is 1 cm.