Background
The part of my project that this comes from is finding the center-point of a circle that passes through three known points, for which I am following the steps in this Mathematics Stack Exchange post. The three points on the circle have the following coordinates:
A = {
r/Sqrt[3]*(Sqrt[2]*Q*(2 - U^2) + 2 - Sqrt[2]/4*c*U^2)/(Q*(2 - U^2) - 2),
r/Sqrt[1]*((c*(Q - 1) - 1)*U) /(Q*(2 - U^2) - 2),
r/Sqrt[6]*(Sqrt[2]*Q*(2 - U^2) + 2 - Sqrt[2]/4*c*U^2)/(Q*(2 - U^2) - 2)
}
B = {
r/Sqrt[3]*(Sqrt[2]*Q*(1 - U^2) + 3 - 1/2*c*(1 + U^2))/(Q*(1 - U^2) - 2),
r/Sqrt[1]*(-(1 + c)*U) /(Q*(1 - U^2) - 2),
r/Sqrt[6]*(Sqrt[2]*Q*(1 - U^2)*(1 + 2*c*Q) + 2*c) /(Q*(1 - U^2) - 2)
}
C = {
r/Sqrt[3]*(Sqrt[2]*Q*(1 - U^2) + 1 + c*(Sqrt[2]*Q*(1 - U^2) + 1))/(Q*(1 - U^2) - 2),
r/Sqrt[1]*(-(1 + c)*U) /(Q*(1 - U^2) - 2),
r/Sqrt[6]*(Sqrt[2]*Q*(1 - U^2) + 4 + c*((1/2 + Q)*(1 - U^2) - 2))/(Q*(1 - U^2) - 2)
}
where r is a positive real number, U is a real number in the interval $[-1, 1]$, c is a real number in the interval $[0, 1]$, and Q := (2 - Sqrt[2])/4.
The example expression in this question comes from the very last step in the linked post and gives the $x$-coordinate of the center-point, Ax + (bx)/2*(ux) + (h)*(vx), where Ax is the $x$-coordinate of point A and where
bx := -2*r*(1 + c*(Q*(1 - U^2) - 1))/Sqrt[1 + 16*(1 - Q)^2 + 2*U^2*(1 + 2*Q^2*(1 + U^2))]
ux := 1/Sqrt[3]*2*(1 + Q*U^2)/Sqrt[1 + 16*(1 - Q)^2 + 2*U^2*(1 + 2*Q^2*(1 + U^2))]
h := -r*(1 + c*(Q*(1 - U^2) - 1))/Sqrt[4*Q^2*(1 + U^2) + 1]
vx := 1/Sqrt[3]*(-8*(1 - Q)^2 - 1/2*U^2)/Sqrt[(1 + 4*Q^2*U^2)*(1 + 16*(1 - Q)^2 + 2*U^2*(1 + 2*Q^2*(1 + U^2)))]
(The above definitions were all simplified using the responses to my previous question.)