Calculate the diameter of an inscribed circle inside a sector of circle $AOB$ is a sector of a circle with center $O$, angle = 45° and radius $OA=10$.
Find the radius of the chord inscribed circle in this sector such that it touches radius $OA$, radius $OB$ and arc $AB$.


The geometry in

says
$$
r=(R-r)\sin\left(\frac\theta2\right)
$$
Therefore, we can solve for $r$:
$$
\bbox[5px,border:2px solid #C0A000]{r=R\,\frac{\sin\left(\frac\theta2\right)}{1+\sin\left(\frac\theta2\right)}}
$$
To find $\theta$ from $R$ and $r$, we can use
$$
\sin\left(\frac\theta2\right)=\frac{r}{R-r}
$$
