Let , , and be the side lengths of a reference triangle . Now let be a point on the extension of the segment beyond such that . Similarly, define the points , , , , so that the points and lie on the extended segment , the points and lie on the extended segment , and the point lies on the extended segment , and we have , , , and .
Then the points , , , , , and are concyclic and the resulting circle is known as Conway circle of .
The Conway circle is centered at the incenter of the reference triangle and has radius
(1)
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(2)
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where is the inradius of the reference triangle and its semiperimeter.
It has circle function
(3)
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corresponding to Kimberling center .
No Kimberling centers lie on the Conway circle.