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.
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
,
.

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Exercise: verify that the side lengths are indeed , , from the coordinates of the vertices.
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