For circle S: x^2+y^2+2gx+2fy+c=0, the value S1 = x1^2+y1^2+2gx1+2fy1+c determines the position of (x1,y1). S1 < 0: inside, S1 = 0: on the circle, S1 > 0: outside. The quantity S1 is called the "power of the point" with respect to the circle. For an external point, the tangent length = sqrt(S1). For a point inside, the minimum chord through it has length related to sqrt(r^2 - d^2) where d is distance from centre.