Find the equation of the circle which is tangent to all four of the circles characterized by these four equations:
x^2+y^2+10x=0
x^2+y^2-10x=0
x^2+y^2+10y=0
x^2+y^2-10y=0
???
$\displaystyle x^2+y^2=100$ should be the required circle.
this is because the radius of each circle is 5 units and their centres are at (-5,0), (5,0), (0,5) and (0,-5).
if you draw a rough figure its easy to see that a symmetric flower shaped digram appears. and then its easy to find the required circle.
In general a circle tangent to four given circles will not exist. In this special symmetric situation it exists.