Dynamic Geometry: P76

Let the centers of the unit, cyan, green, and purple semicircles be $O$ , $A$ , $B$ , and $C$ , their radii $1$ , $r_1$ , $r_2$ , and $r$ , Then $r_1 + r_2 = 1$ , $OA = 1-r_1 = r_2$ , and $OB = 1-r_2 = r_1$ .

Let diameters $DE$ and $FG$ through centers $C$ and $O$ respectively. By intersecting chords theorem ,

$\begin{aligned} DC\cdot CE & = FC \cdot CG \\ r^2 & = (1-k)(1+k) = 1 - k^2 \\ \implies k^2 & = 1 - r^2 \end{aligned}$

Let $\angle COB = \theta$ . By cosine rule ,

$\begin{cases} BC^2 = OB^2 + OC^2 - 2 \cdot OB \cdot OC \cdot \cos \theta \\ CA^2 = OA^2 + OC^2 + 2 \cdot OA \cdot OC \cdot \cos \theta \end{cases} \\ \begin{cases} (r_2+r)^2 = r_1^2 + (1-r^2) - 2r_1\sqrt{1-r^2} \cos \theta & ...(1) \\ (r_1+r)^2 = r_2^2 + (1-r^2) + 2r_2\sqrt{1-r^2} \cos \theta & ...(2) \end{cases} \\ \begin{aligned} r_2 \times (1) + r_1 \times (2): \quad r_1^3+r_2^3 + 2(r_1^2+r_2^2)r + \blue{(r_1+r_2)} r^2 & = r_1r_2 \blue{(r_1+r_2)} +\blue{(r_1+r_2)}(1-r^2) \\ \blue{(r_1+r_2)}(\blue{(r_1+r_2)}^2-3r_1r_2) + 2(\blue{(r_1+r_2)}^2 - 2r_1r_2)r + r^2 & = r_1r_2 + 1 - r^2 \quad \quad \small \blue{\text{Note that }r_1+r_2 = 1} \\ 1 - 3r_1r_2 + 2r - 4r_1r_2r + r^2 & = r_1r_2 + 1 - r^2 \\ r + r^2 & = 2r_1r_2 + 2r_1r_2r \\ \implies r & = 2r_1r_2 \end{aligned}$

Putting $r = \dfrac {56}{121}$ , then we have:

$\begin{aligned} 2r_1r_2 & = \frac {56}{121} & \small \blue{\text{Note that }r_2 = 1-r_1} \\ r_1 (1-r_1) & = \frac {28}{121} \\ 121 r_1^2 - 121r_1 + 28 & = 0 \\ (11r_1 - 4)(11r_1 - 7) & = 0 & \small \blue{\text{Since }r_1 \text{ and }r_2 \text{ are interchangeable,}} \\ \implies r_1, r_2 & = \frac 4{11}, \frac 7{11} \end{aligned}$

Then the sum of areas of the cyan and green semicircles $\dfrac \pi 2 \left(\left(\dfrac 4{11}\right)^2 + \left(\dfrac 7{11} \right)^2 \right) = \dfrac {65}{242}$ . Therefore $p+q = 65+242 = \boxed{307}$ .