Radical Centre of Three Circles

Now consider three circles $S_1=0,S_2=0,S_3=0$ whose centres are not all on one line. Pairing them up gives three radical axes: that of $S_1,S_2$ (namely $L_{12}=S_1-S_2$), of $S_2,S_3$ ($L_{23}=S_2-S_3$), and of $S_3,S_1$ ($L_{31}=S_3-S_1$).

These three linear expressions satisfy an identity that makes their geometric relationship immediate:

Because the three equations sum identically to the zero polynomial, any point that satisfies two of them automatically satisfies the third. So the three radical axes are concurrent: all three pass through one common point, called the radical centre of the three circles. (If the three centres happen to be collinear, the three radical axes -- each perpendicular to a segment between two of the collinear centres, hence all mutually parallel -- fail to meet in a single point at all, unless they in fact coincide.) …