Circle

This is an interesting problem......

Let R be the radius of the larger circle  and r  the radius of the smaller circle

Note  that  F = (0, R)   and   H = (R,0)

FH =   sqrt ( R^2 + R^2 )  =  R*sqrt (2)

Likewise BC = r*sqrt (2)

Midpoint of  FH  =  (R/2  R/2)

Distance  from O  to the midpoint =  R/sqrt (2)

Midpoint of BC =   (r/2, r,2) 

Distance  from O to this midpoint  = r/sqrt (2)

The  quadrilateral is a  trapezoid  with  height =  (R + r) / sqrt (2)

And  the sum of its bases =  sqrt (2) (R + r)

Area of  quadrilateral =   (1/2) (R + r)  / sqrt (2)  *  ( sqrt (2) (R + r) )  =  (1/2) ( R + r)^2

Area  between the circles  =  pi (R^2  - r^2)

Area of quadrilateral / area between circles =   2/3

So

(1/2) (R + r)^2             2

___________  =       ____

pi (R^2 - r^2)               3

(1/2) (R + r) (R + r)           2

_______________  =      ___

pi ( R + r) ( R - r)               3

(1/2) (R + r)                2

_________  =         ________      cross-multiply

pi (R - r)                      3

(3/2) ( R + r)  = 2pi ( R - r)

(3/2)R  + (3/2)r  =  2pi R  - 2pi r

( 2pi + 3/2  ) r =  (2pi - 3/2) R

r / R  =  (2pi - 3/2)  / (  2pi + 3/2 ) ≈  .6145

Large circle radius R = ?        Small circle radius r = 1

Large circle area            A_l = R²pi           Small circle area      A_s = pi

Trapezoid area               A_t = R²/2 + R + 1/2

Annular ring area           A_r = R²pi - pi

R²pi - pi  / (R²/2 + R + 1/2) = 3/2

R = 1.62719711 units

So, the ratio is      1 : 1.62719711       or        0.614553697