$[$ Properties and characteristics of the tangent $]$

In a circle with radius $R=4$, a chord $A B$ and a diameter $A K$ are drawn, forming an angle of $\frac{\pi}{8}$ with the chord. A tangent is drawn at point $B$, intersecting the extension of the diameter $A K$ at point $C$. Find the median $A M$ of triangle $A B C$.
Let $O$ be the center of the circle. Then $O B C$ is an isosceles right triangle.

## Solution

Let $O$ be the center of the circle. Then

$$
\angle B O C=2 \angle B A O=45^{\circ}
$$

From the right triangle $O B C$, we find that $B C=4$ and $O C=4 \sqrt{2}$. Therefore, $A C=A O+O C=4+4 \sqrt{2}$.

We find the median $A M$ using the cosine rule in triangle $A M C$:

$$
\begin{gathered}
A M^{2}=A C^{2}+C M^{2}-2 A C \cdot C M \cos 45^{\circ}= \\
=(4+4 \sqrt{2})^{2}+4-2 \cdot 2 \cdot(4+4 \sqrt{2}) \cdot \frac{\sqrt{2}}{2}=4(9+6 \sqrt{2})
\end{gathered}
$$

![](https://cdn.mathpix.com/cropped/2024_05_06_16ab69fab1743849dba9g-20.jpg?height=389&width=646&top_left_y=1875&top_left_x=706)

## Answer

$2 \sqrt{9+6 \sqrt{2}}$.