The general equation x^{2+y}^{2+2gx+2fy+c}=0 represents a circle with centre (-g,-f) and radius sqrt(g^{2+f}^{2-c}). For a valid real circle, g^{2+f}^{2-c} must be positive. When it equals zero, we get a point circle; when negative, the circle is imaginary. The equation must have equal coefficients of $x^2$ and $y^2$ and no xy term. The diameter form (x-x1)(x-x2)+(y-y1)(y-y2)=0 is useful when two diametrically opposite points are known, exploiting the fact that the angle in a semicircle is 90 degrees. The parametric representation x=h+rcos(theta), y=k+rsin(theta) traces all points on the circle. The power of a point S1=x1^{2+y1}^{2+2gx1+2fy1+c} determines the position: S1<0 (inside), S1=0 (on), S1>0 (outside). The tangent length from an external point equals sqrt(S1), the x-axis intercept is 2sqrt($g^{2-c}$), and the y-axis intercept is 2sqrt($f^{2-c}$).