A quarter-circle with radius is drawn. A circle is drawn inside the sector, which is tangent to the sides of the sector, as shown. Find the radius of the inscribed circle.
Call R the radius of the larger quarter-circle and r be the radius of the smaller circle
Let P be the center of the smaller circle
And we can show that the distance from O to P = √2r
R = r + √2 r
R = r ( 1 + √2)
r = R / ( 1 + √2)
Let P be the point of tangency of the small circle with line segment AO.
Let Q be the point of tangency of the small circle with line segment BO.
Let R be the center of the small circle.
OPRQ is a square.
Let the distance from O to P be x.
Then the radius of the small circle is also x.
Since OPRQ is a square with each side x, the distance from O to R is x·sqrt(x).
Then the radius of the large circle (with center O and radius OA) is x + x·sqrt(x).
Now, if you know the radius of the large circle, you can find the radius of the small circle.