Dynamic Geometry: P51

Let the center of the semicircle be $O(0,0)$ , the origin of the $xy$ -plane, the triangle be $ABC$ , where $AB$ is the diameter of the semicircle and $C(u,v)$ , the red point, and $P(x,y)$ , the centroid. Since $P$ is the centroid of $\triangle ABC$ , its coordinates are given by:

$\begin{cases} x = \dfrac {x_A+x_B+x_C}3 = \dfrac {-1+1+u}3 & = \dfrac u3 \\ y = \dfrac {y_A+y_B+y_C}3 = \dfrac {0+0+v}3 & = \dfrac v3 \end{cases}$

Since the locus of $C(u,v)$ is a semicircle with radius $1$ , the locus $P(x,y)$ is also a semicircle with a smaller radius of $\dfrac 13$ . Therefore the area bounded by the locus and the cyan segment is $A = \dfrac 12 \cdot \pi \left(\dfrac 13 \right)^2 = \dfrac \pi{18}$ . Therefore $a = \boxed{18}$ .