If the circumference of a circle with radius $\mathrm{R}$ is equal to the sum of the circumferences of two circles with radii $R_1$ and $R_2$ then:

The circumference of a circle with radius ${\mathrm{R}}_1=2{\mathrm{\pi R}}_1$, and

the circumference of circle with radius ${\mathrm{R}}_2=2{\mathrm{\pi R}}_2$,

Therefore the sum of the above circumferences would be:

 $Sum=2\mathrm{\pi }\left({\mathrm{R}}_1+{\mathrm{R}}_2\right)$.

Now, the circumference of circle with radius $\mathrm{R}=2\mathrm{\pi R}$.

According to the given condition we get:

$2\mathrm{\pi }\left({\mathrm{R}}_1+{\mathrm{R}}_2\right)=2\mathrm{\pi R}$

$\Rightarrow {\mathrm{R}}_1+{\mathrm{R}}_2=\mathrm{R}$