Radius of third touching circle

Let the two equal circles have radius $R = 4\text{ cm}$ and the third circle have radius $r\text{ cm}$.

The length of a direct common tangent segment between two externally touching circles of radii $r_1$ and $r_2$ is $2\sqrt{r_1 r_2}$.

The distance between the points of contact on the common tangent for the two large circles is $2\sqrt{4 \times 4} = 8$. This distance is also the sum of the direct common tangent segments from each large circle to the small circle: $2\sqrt{4r} + 2\sqrt{4r} = 8$ $4\sqrt{r} + 4\sqrt{r} = 8 \implies 8\sqrt{r} = 8 \implies \sqrt{r} = 1 \implies r = 1\text{ cm}$