I've been using Wolfram Engine 14.3 for a personal project, one part of which required finding the center-point of a circle in 3D space when given the coordinate-triples of three points on the circle's boundary (see below). I followed the steps in this Mathematics Stack Exchange post but wound up with extremely long expressions that were tricky to simplify (as referenced in my first question about simplification here and my second question about simplification here). The user @azerbajdzan suggested I post a question specifically about finding the center-point as they had a better way to solve it that did not require all the tedious simplification steps.
The three points on the circle have the following coordinates:
pointA = {
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)
}
pointB = {
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)
}
pointC = {
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.
