Dynamic Geometry: P27

Let the center of the semicircle be the origin $O(0,0)$ of the $xy$ -plane with the base of the rectangle be the $x$ -axis. Since the figure is symmetrical about the $y$ -axis, let us just consider only the positive quadrant. Let any point on the locus or the center of any circle satisfying the conditions be $P(x,y)$ , when the radius of the circle is $r$ . Then we note that:

$\begin{cases} x = 1 - r & \implies r = 1-x \\ y = \sqrt{(1+r)^2-x^2} = \sqrt{(2-x)^2-x^2} & \implies y^2 = 4 - 4x \end{cases}$

We note that the red curve $y^2 = 4-4x$ is a parabola. The area bounded by the parabola and two axes is $\dfrac 23$ the area of the smallest rectangle with the two axes as sides that inscribes the parabola. In this case the rectangle has vertices at $(0,0)$ , $(0,2)$ , $(1,2)$ , and $(1,0)$ , whose area is $2 \times 1 = 2$ . Therefore the area bounded by the two red curves and the $x$ -axis is $2 \times 2 \times \dfrac 23 = \dfrac 83$ and the area bound by the two red curves and the semicircle is $\dfrac 83 - \dfrac \pi 2 = \dfrac {16-3\pi}6$ . The answer to $\sqrt{a+b+c} = \boxed 5$ .