know that the radical axis of two circles is the best alternative line to the line connecting the intersection points of two non-intersecting circles
in the same way I'm trying to find the best alternative circle for the circle passing through the vertices of a noncircular quadrilateral
I'm looking for an actual geometric alternative that has common theorems with the circle passing through the vertices of a ring quadrilateral, so I'm not looking for naïve answers like the smallest possible circle with the four points inside it or the largest possible circle that makes the four points outside it or other answers like that
Two days ago, I met a theorem that brought me closer to reaching the answer I am looking for, as the theorem helps to find the right point to be the center of that circle we are looking for
If $ABCD$ quadrilateral circular, $AB, CD$ They intersect in $M$ and $BC, AD$ They intersect in $N$ and $AC, BD$ They intersect in $P$, the point of convergence of heights in the triangle $MNP$ is the center of the circle passing through the vertices of the quadrilateral $ABCD$.
This theorem is known as the Brookard theorem and obviously its idea can be applied for any quadrilateral, even if it's not a circular quadrilateral, so I think I found the center of the circle I'm looking for.
But I haven't found any theorem that indicates the radius, is there anyone who can find the radius of this circle?
And I'd be happy if someone could find more common properties between the prospective circle and the circle in the case of a circular quadrilateral.
I hope my question is clear
