First we note that the number of points of intersection of the lines is $k \cdot m$ (figure 1).

Figure 1

Let now all the points of intersection lie at the exterior of the circle with the greatest radius. Then the $k$ different lines passing through $A$ create $2k$ regions inside the circle with the smaller radius, while the $m$ different lines passing through $B$ create $2m$ regions inside the circle with the smaller radius. Since we have counted the shaded region (figure 2) two times, the total number of regions inside the smaller circle is $2 \cdot k + 2 \cdot m - 1$.

We have totally $n-1$ circular rings. In each circular ring is created the same number of regions $2 \cdot k + 2 \cdot m - 1$, but the shaded region is divided in two regions (figures 2α and 2β). Therefore inside any circular ring we have $2 \cdot k + 2 \cdot m$ regions.

figure 2β

figure 2α

Finally in this case the total number of regions is: $2n(k+m)-1$.

If one or more points of intersection lie inside the circle with the smaller radius, then we have $2(k+m)$ regions inside the smaller circle (figures 3α, 3β και 3γ). In the case of figure 3α (all the points of intersection lie inside the circle with the smaller radius) we have $2(n-1)(k+m)$ regions into the $n-1$ circular rings. In the case one point of intersection lie into a circular ring the number of region increases by one. Therefore, finally we have totally
$$2(k+m)+2(n-1)(k+m)=2n(k+m) \text{ regions.}$$

figure 3α