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The radical circle of three circles has center at the radical center and is orthogonal to each of the three circles.

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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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Contributed by: Minh Trinh Xuan (August 2022)
Open content licensed under CC BY-NC-SA

Details

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

Permanent Citation

Minh Trinh Xuan

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