Suppose four points \(A,B,C,D\) lie on a common circle. How do the four quadrances to the point \(M:\equiv(AC)(BD)\) relate?
Let me Gröbner that for you.
% sage ┌──────────────────────┐ │ SageMath version 8.8 │ └──────────────────────┘ sage: R.<x0,y0,x1,y1,x2,y2,x3,y3,s,t,q0,q1,q2,q3> = PolynomialRing(QQ, order='lex') sage: xm, ym, xm1, ym1 = x0*s+x2*(1-s), y0*s+y2*(1-s), x1*t+x3*(1-t), y1*t+y3*(1-t)
sage: ideal(x0^2+y0^2-1, x1^2+y1^2-1, x2^2+y2^2-1, x3^2+y3^2-1,
....: xm-xm1, ym-ym1,
....: -q0+(xm-x0)^2+(ym-y0)^2, -q1+(xm-x1)^2+(ym-y1)^2,
....: -q2+(xm-x2)^2+(ym-y2)^2, -q3+(xm-x3)^2+(ym-y3)^2).groebner_basis()[-1]
q0*q2 - q1*q3\[Q(A,M)Q(C,M)=Q(B,M)Q(D,M)\]
