Calculate the diameter of an inscribed circle inside a sector of circle

Question

$AOB$ 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 $AB$.

Answer 1

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 $$ \sin\left(\frac\theta2\right)=\frac{r}{R-r} $$

Answer 2

If $\theta$ is the sector angle, a coordinate geometry solution would be to consider a circle $x^2 + y^2 + 2gx + 2fy + c =0 $ inside the circle $x^2 + y^2 = 100$. Now our required circle touches lines $y=0$, $y=\tan \theta x $ and the circle. I suppose you can do it now. For tangency to lines, distance of centre from line is equal to radius, add for touching circle, distance between centres = difference of radii.