B4. In the figure, you see a 'pointed arch' $A B C$ and its inscribed circle. The pointed arch consists of line segment $A B$ with length 1, circular arc $B C$ with center $A$, and circular arc $A C$ with center $B$.

How large is the radius of the inscribed circle of the pointed arch?

![](https://cdn.mathpix.com/cropped/2024_04_17_a91c336121b9423a3a1eg-2.jpg?height=297&width=320&top_left_y=2373&top_left_x=1553)
B4. Consider the inscribed circle. Let its center be $O$ and the length of its radius be $r$. The point of tangency with $A B$ is called $M$ and the point of tangency with arc $B C$ is called $R$.

The three points $A, O$, and $R$ lie on a line, so $|A O|=|A R|-|O R|=$ $1-r$. Furthermore, $|O M|=r$ and $|A M|=\frac{1}{2}|A B|=\frac{1}{2}$. The Pythagorean theorem gives $|A M|^{2}+|O M|^{2}=|A O|^{2}$, and thus $\frac{1}{4}+r^{2}=(1-r)^{2}=$

![](https://cdn.mathpix.com/cropped/2024_04_17_7effb8aeb7220f8b3452g-2.jpg?height=297&width=323&top_left_y=2287&top_left_x=1546)
$r^{2}-2 r+1$. From this, it follows that $2 r=\frac{3}{4}$ and thus $r=\frac{3}{8}$.