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Stammler Circles Radical Circle


StammlerCirclesRadicalCircle

The radical circle of the Stammler circles has center at the nine-point center N, which is Kimberling center X_5. The radius is given by

R_S=sqrt(R^2+ON^2)
(1)
=1/2sqrt(4R^2+OH^2)
(2)
=(sqrt(a^6-b^2a^4-c^2a^4-b^4a^2-c^4a^2+7b^2c^2a^2+b^6+c^6-b^2c^4-b^4c^2))/(8Delta)
(3)

(P. Moses and J.-P. Ehrmann, pers. comm., Jan. 28, 2004), where R is the circumradius, O is the circumcenter, H is the orthocenter, N is the nine-point center, and Delta is the area of the reference triangle.

Its circle function is given by

 l=(b^6-2a^2b^4-c^2b^4+a^4b^2-c^4b^2+2a^2c^2b^2+c^6-2a^2c^4+a^4c^2)/(32Delta^2bc),
(4)

which corresponds to Kimberling center X_(3003).

No Kimberling centers lie on it.

StammlerCirclesRadicalCircleRadicalAxis

The radical line of the circumcircle and Stammler circles radical circle passes through the circumcenter O (i.e., bisects the circumcircle) and is perpendicular to the Euler line (P. Moses, pers. comm., Jan. 28, 2005).

StammlerCirclesRadicalCircleRadicalAxis2

The radical line of the Stammler circle and Stammler circles radical circle passes through the Kimberling center X_(23) and is perpendicular to the Euler line (P. Moses, pers. comm., Jan. 28, 2005).

StammlerCirclesRadicalCircleRadicalAxis3

The radical line of the first Droz-Farny circle and the Stammler circles radical circle passes through X_(403), perpendicular to the Euler line and the radical axis of the second Droz-Farny circle and the Stammler radical circle passes through X_(2072), also perpendicular to the Euler line (P. Moses, pers. comm., Jan. 28, 2005).


See also

Radical Circle, Stammler Circles

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Cite this as:

Weisstein, Eric W. "Stammler Circles Radical Circle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/StammlerCirclesRadicalCircle.html

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