Exercise 5. 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 $C D$.
## Solution to Exercise 5

![](https://cdn.mathpix.com/cropped/2024_06_04_b425935e33d3e5c20a22g-2.jpg?height=360&width=661&top_left_y=514&top_left_x=729)

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 $C D = \sqrt{9 - 32 / 9} = 7 / 3$.