Proof of a circle defined by five points in projective geometry

Question

I'm trying to understand how to prove that a circle can be defined in projective geometry by five points, where three are "classic points" and two are the circular points. For demostrating that is definable with three points I'm starting by the equation of a generic conic that has 5 DOF, and I reduce it to the equation of a circle showing that has 3 DOF, confirming it putting in matrix form showing that the circle is constrained by three equations. But how can I continue to show for the circular points?

Answer 1

Too long for a comment:

$$\det\begin{pmatrix} x^2 &x\,y&y^2&x&y &1\cr x_1^2&x_1\, y_1&y_1^2&x_1 &y_1&1\cr x_2^2 &x_2\,y_2&y_2^2 &x_2&y_2&1 \cr x_3^2&x_3\,y_3 &y_3^2&x_3&\ y_3&1\cr 1&i&-1 &0&0&0\cr 1&-i &-1&0&0&0\cr \end{pmatrix}=2\,i\,\left(x_1\,y_2\,y_3^2-x\,y_2 \,y_3^2-x_2\,y_1\,y_3^2+x\, y_1\,y_3^2+x_2\,y\,y_3^2-x_1 \,y\,y_3^2-x_1\,y_2^2\,y_3+x\, y_2^2\,y_3+x_2\,y_1^2\,y_3-x \,y_1^2\,y_3-x_2\,y^2\,y_3+ x_1\,y^2\,y_3-x_1\,x_2^2\, y_3+x\,x_2^2\,y_3+x_1^2\,x_2 \,y_3-x^2\,x_2\,y_3-x\,x_1^2\, y_3+x^2\,x_1\,y_3+x_3\,y_1\, y_2^2-x\,y_1\,y_2^2-x_3\,y\, y_2^2+x_1\,y\,y_2^2-x_3\,y_1 ^2\,y_2+x\,y_1^2\,y_2+x_3\,y^2\, y_2-x_1\,y^2\,y_2+x_1\,x_3^2 \,y_2-x\,x_3^2\,y_2-x_1^2\, x_3\,y_2+x^2\,x_3\,y_2+x\, x_1^2\,y_2-x^2\,x_1\,y_2+x_3 \,y\,y_1^2-x_2\,y\,y_1^2-x_3\,y^2\, y_1+x_2\,y^2\,y_1-x_2\,x_3^2 \,y_1+x\,x_3^2\,y_1+x_2^2\, x_3\,y_1-x^2\,x_3\,y_1-x\, x_2^2\,y_1+x^2\,x_2\,y_1+x_2 \,x_3^2\,y-x_1\,x_3^2\,y-x_2^2\, x_3\,y+x_1^2\,x_3\,y+x_1\, x_2^2\,y-x_1^2\,x_2\,y\right)=0$$

$$\det\begin{pmatrix} y^2+x^2 &x&y&1\cr y_1^2+ x_1^2&x_1&y_1 &1\cr y_2^2+x_2^2& x_2&y_2&1\cr y_3^2 +x_3^2&x_3&y_3 &1\cr \end{pmatrix}=-\left(x_1\,y_2\,y_3^2-x\,y_2\, y_3^2-x_2\,y_1\,y_3^2+x\,y_1 \,y_3^2+x_2\,y\,y_3^2-x_1\,y\, y_3^2-x_1\,y_2^2\,y_3+x\,y_2 ^2\,y_3+x_2\,y_1^2\,y_3-x\, y_1^2\,y_3-x_2\,y^2\,y_3+x_1 \,y^2\,y_3-x_1\,x_2^2\,y_3+x\, x_2^2\,y_3+x_1^2\,x_2\,y_3-x ^2\,x_2\,y_3-x\,x_1^2\,y_3+x^2\, x_1\,y_3+x_3\,y_1\,y_2^2-x\, y_1\,y_2^2-x_3\,y\,y_2^2+x_1 \,y\,y_2^2-x_3\,y_1^2\,y_2+x\, y_1^2\,y_2+x_3\,y^2\,y_2-x_1 \,y^2\,y_2+x_1\,x_3^2\,y_2-x\, x_3^2\,y_2-x_1^2\,x_3\,y_2+x ^2\,x_3\,y_2+x\,x_1^2\,y_2-x^2\, x_1\,y_2+x_3\,y\,y_1^2-x_2\, y\,y_1^2-x_3\,y^2\,y_1+x_2\,y^2\, y_1-x_2\,x_3^2\,y_1+x\,x_3^2 \,y_1+x_2^2\,x_3\,y_1-x^2\, x_3\,y_1-x\,x_2^2\,y_1+x^2\, x_2\,y_1+x_2\,x_3^2\,y-x_1\, x_3^2\,y-x_2^2\,x_3\,y+x_1^2\, x_3\,y+x_1\,x_2^2\,y-x_1^2\, x_2\,y\right)=0$$