10. As shown in the figure, two circles are placed side by side on a straight line, touching each other. The area of each circle is 1680 square centimeters. The shaded area is the region between the two circles and the straight line. If a circle of the largest possible size is to be placed inside the shaded area, what is the area of this circle in square centimeters?
【Solution】Solution: As shown in the figure,
$A, B, C$ are the centers of 3 circles,
Let the radius of the large circle be $R$ cm, and the radius of the small circle be $r$ cm, then $A C=\mathbf{R}+r$ (cm) $C D=\mathbf{R}-r$ (cm);

Since $A D^{2}+C D^{2}=A C^{2}$,
Therefore, $R^{2}+(\mathbf{R}-r)^{2}=(\mathbf{R}+r)^{2}$,
Rearranging, we get $R=4 r$,
So $R^{2}=16 r^{2}$,
So $\pi R^{2}=16 \pi r^{2}$,
So $\pi r^{2}$
$=\pi R^{2} \div 16$
$=1680 \div 16$
$=105$ (square cm)
Answer: The area of this circle is 105 square cm.