On a semicircle with diameter $[A D]$, two points $B$ and $C$ are placed such that $A B=B C=1$. Suppose that $A D=3$. Calculate the length $C D$.

## - Statements intended for high school students -
Let $H$ be the intersection of $[A C]$ with $[O B]$, where $O$ is the center of the circle. We have $C D = \sqrt{A D^{2} - A C^{2}}$ with $A C = 2 A H$.

Let $K$ be the midpoint of $[A B]$. The triangle $A O B$ is isosceles at $O$. We denote $\alpha = \widehat{H A B} = \widehat{B O K}$. We have $\cos \alpha = A H / A B = O K / O B$ so $A H = \frac{2}{3} O K$. Furthermore, $O K = \sqrt{O A^{2} - A K^{2}} = \sqrt{(3 / 2)^{2} - (1 / 2)^{2}} = \sqrt{2}$, thus $A C = \frac{4}{3} \sqrt{2}$, and finally $C D = \sqrt{9 - 32 / 9} = 7 / 3$.