The common chords of any three circles

        Theorem     The common chords of any three circles are concurrent.

       

        Proof

 

                Here is an interesting proof using the         theory of determinant.

 

        Let the equations of any three circles be:

 

       

       

        The three common chords are :

               

        We can then form the coefficient determinant using these equations:

               

 

                                                                                  =  0  (since the entries of row 3 are zero)

 

        \      are concurrent.

 

 

        Note

 

                This theorem is also true if the circles are not intersecting. In such case, the radical axes have         to be used instead of the common chords.