### Theory [Equation of a circle passing through the intersection points of two circles and another point]

Consider that the equation of the two circles are \(g(x,y)\) and \(f(x,y)\). The third circle passes through the Intersecting points of these two circles, whose equation can be derived according to A. R. Khalifa Rule, which is: $$p(x,y)=g(x,y)+k.f(x,y)=0$$ The circle \(p(x,y)\) passes through an external point \((m,n)\) upon which none of the circles \(g(x,y)\) and \(f(x,y)\) passes through. Thus, we find \(k\): $$k=-\frac{g(m,n)}{f(m,n)}$$ Putting this k back into the former equation, we find our circle.