Notes
nested semi-circles solution

Nested Semi-Circles

Solution by Chord Properties and Pythagoras' Theorem

In the above diagram, $O$ is the midpoint of $A B$ and so is the centre of the smallest semi-circle while $Q$ is the midpoint of $C D$ and so is the centre of the middle semi-circle.

As $A B$ is a chord of the middle semi-circle, the perpendicular bisector of $A B$ passes through its centre. That bisector passes through the midpoint, which is $O$, and so $O Q$ is perpendicular to $A B$. The line segment $C D$ is tangential to the smallest circle and so is perpendicular to $O Q$ since the angle between a radius and tangent is a right angle. This means that $O Q$ is also the perpendicular bisector of $C D$ and so $O$ is also the centre of the largest semi-circle.

Let $a$, $b$, $c$ be the radii of the three semi-circles in increasing order. Applying Pythagoras' theorem to triangle $Q O A$ shows that $b^2 = a^2 + a^2 = 2 a^2$. Applying it to triangle $O Q C$ shows that $c^2 = b^2 + a^2 = 3 a^2$.

The area of the largest semi-circle is therefore $3$ times that of the smallest, so $\frac{1}{3}$ of the largest is shaded.