circle that touch quadrantal internally

Question

I want to know how to construct circle that touch quadrantal(1/4 part of circle) internally.

I spend several hours for solving this problem but I have no luck.

I attached the picture what I've tried

could somebody please tell me how to construct step by step?

thanks in advanced

Answer 1

Construct the tangent line of the quadrantal at the midpoint of it's arc. Intersect that with the radial lines of the quadrantal. Since the inner circle must be tangent to both the outer circle and the constructed tangent at the same point, the problem is reduced to finding the inscribed circle of the triangle.

The inscribed circle of the triangle has a center at the intersection of the angle bisectors of the triangle.