Assume without loss of generality that x = 1 unit. Scale for higher values of X.

(Currently researching the problem, and equations. Maybe numeric solution???)

Let 2*theta be the radial angle of the sqrt(2) chord.
Let 2*phi be the radial angle of the 1 unit chord.
2*theta + 8*phi = 360 degrees = 2*PI
theta = PI - 4*theta
Let r = radius of circle.
sin(theta) = sqrt(2)/2r = 1/(sqrt(2)*r)
sin(phi) = 0.5/r = 1/(2*r)