Radical Circle of Three Circles
The radical circle of three circles has center at the radical center and is orthogonal to each of the three circles.[more]
Let the centers of the circles form the triangle and let its side lengths be , , .
Using Conway triangle notation, define
Half the triangle's area is then given by
In those terms, suppose the Cartesian coordinates of three centers are at , , and the corresponding radii are , , .
Then the radical circle of the three circles , , is the circle , where
Exercise: verify that the side lengths are indeed , , from the coordinates of the vertices.