Given the two points (0,0) and (xg,yg) [the things in red].
The lower circle's origin is on the x-axis and it is tangent to the y-axis at 180 deg (it's left-most point).
The upper circle's origin is on y=yg and it is tangent to (xg,yg) at 0 deg (it's right-most point).
The two circles must have a tangent point (xt,yt) such that:
0 < xt < xg and 0 < yt < yg
(that is, the tangent point must be within the rectangle in the picture).
Solving for an equation with which to calculate the radii of the two circles and their tangent point [the things in blue].
Thank you so much for your help!