(14) (16 points) As shown in the figure, in a certain engineering project to measure the radius $R$ of an arc, three identical small balls are placed on the arc so that each contact point is tangent to the arc. The height difference between the balls is $h$, and the radius of the small balls is $r$. Try to express $R$ in terms of $h$ and $r$, and find the value of $R$ when $r=100$ and $h=40$.
14 Let the angle between the line connecting the centers of two tangent small circles and the line connecting the center of the large circle be $\theta$, then
$$
\cos \theta=\frac{(R-r)^{2}+(R-r)^{2}-(2 r)^{2}}{2(R-r)^{2}} .
$$

Also, $h=(R-r)-(R-r) \cos \theta=\frac{2 r^{2}}{R-r}$, so, $R=r+\frac{2 r^{2}}{h}$.
When $r=100, h=40$, $R=100+\frac{2 \times 100^{2}}{40}=600$.