Notes
three semi-circles solution

Three Semi-Circles

Solution by Pythagoras' Theorem and Circle Area

Let $a$, $b$, $c$ be the radii of the three semi-circles in increasing order. With the points labelled as above, then $a$ is the length of $B F$, $b$ of $A D$ and $D E$, and $c$ of $C E$. Each of $A B$, $B C$, and $C D$ are one third of the radius of the middle semi-circle, so are $\frac{b}{3}$.

Applying Pythagoras' theorem to triangle $F B D$ gives the relationship:

so $\frac{5}{9} b^2 = a^2$.

Applying Pythagoras' theorem to triangle $C D E$ gives the relationship:

so $c^2 = \frac{10}{9} b^2 = 2 a^2$.

The area of the larger semi-circle is $\frac{1}{2} \pi c^2 = \pi a^2$. The area of the smallest semi-circle is $\frac{1}{2} \pi a^2$, which is $5$, so the largest semi-circle has area $10$.