Circles $A, B,$ and $C$ each have radius 1. Circles $A$ and $B$ share one point of tangency. Circle $C$ has a point of tangency with the midpoint of $\overline{AB}.$ What is the area inside circle $C$ but outside circle $A$ and circle $B?$ $\textbf{(A)}\ 3 - \frac{\pi}{2} \qquad$ $ \textbf{(B)}\ \frac{\pi}{2} \qquad$ $ \textbf{(C)}\ 2 \qquad$ $ \textbf{(D)}\ \frac{3\pi}{4} \qquad$ $ \textbf{(E)}\ 1+\frac{\pi}{2}$
Solution
Place $A=(-1,0), B=(1,0)$ (tangent at the origin $M$), and since $C$ is tangent to line $AB$ at $M$ with radius $1$, its center is $(0,1)$.
The distance from $C$'s center to $A$'s center is $\sqrt{1^2+1^2}=\sqrt2$; by symmetry the same holds for $B$.
For two unit circles with centers distance $d$ apart, the overlap area is $2\cos^{-1}\left(\frac d2\right)-\frac d2\sqrt{4-d^2}$. With $d=\sqrt2$: overlap $=2\cos^{-1}\left(\frac{\sqrt2}2\right)-\frac{\sqrt2}2\sqrt{2}=2\cdot\frac\pi4-1=\frac\pi2-1$.
Since $A$ and $B$ meet circle $C$ only pairwise (their own tangency point lies on $C$'s boundary, contributing no area), the desired area is $$[C]-2\left(\frac\pi2-1\right)=\pi-\pi+2=\boxed2.$$
Circles $A,B$ each have radius $1$ and share a point of tangency. Circle $C$ has radius $\sqrt3$ and is tangent to line $AB$ at the midpoint of $\overline{AB}$. What is the area inside circle $C$ but outside circles $A$ and $B$?
$\textbf{(A)}\ 3 - \frac{\pi}{2} \qquad$ $ \textbf{(B)}\ \frac{\pi}{2} \qquad$ $ \textbf{(C)}\ 2 \qquad$ $ \textbf{(D)}\ \frac{3\pi}{4} \qquad$ $ \textbf{(E)}\ 1+\frac{\pi}{2}$ 
