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Conway Circle


ConwayCircle

Let a, b, and c be the side lengths of a reference triangle DeltaABC. Now let A_b be a point on the extension of the segment CA beyond A such that AA_b=a. Similarly, define the points A_c, B_c, B_a, C_a, C_b so that the points A_c and B_c lie on the extended segment AB, the points B_a and C_a lie on the extended segment BC, and the point C_b lies on the extended segment CA, and we have AA_c=a, BB_c=b, BB_a=b, CC_a=c and CC_b=c.

Then the points A_b, A_c, B_c, B_a, C_a, and C_b are concyclic and the resulting circle is known as Conway circle of DeltaABC.

The Conway circle is centered at the incenter I of the reference triangle DeltaABC and has radius

R_C=sqrt(r^2+s^2)
(1)
=sqrt((a^2b+ab^2+a^2c+abc+b^2c+ac^2+bc^2)/(abc)),
(2)

where r is the inradius of the reference triangle and s its semiperimeter.

It has circle function

 l=(a(b+c))/(bc),
(3)

corresponding to Kimberling center X_(213).

No Kimberling centers lie on the Conway circle.


See also

Central Circle, Trilinear Coordinates

Portions of this entry contributed by Darij Grinberg

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Cite this as:

Grinberg, Darij and Weisstein, Eric W. "Conway Circle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ConwayCircle.html

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