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Second Steiner Circle


SecondSteinerCircle

The second Steiner circle (a term coined here for the first time) is the circumcircle of the Steiner triangle DeltaS_AS_BS_C.

Its center has center function

 alpha=-((b^2-c^2)f(a,b,c))/a,
(1)

where

 f(a,b,c)=a^8-4b^2a^6-4c^2a^6+6b^4a^4+6c^4a^4+b^2c^2a^4-4b^6a^2-4c^6a^2+b^2c^4a^2+b^4c^2a^2+b^8+c^8-2b^2c^6+2b^4c^4-2b^6c^2,
(2)

which is not a Kimberling center. The radius is given by

 R_S=(sqrt(f(a,b,c)f(b,c,a)f(c,a,b)))/(2abc|(a^2-b^2)(a^2-c^2)(b^2-c^2)|)R,
(3)

where

 f(a,b,c)=b^6+c^6-a^6+3a^4b^2-3a^2b^4+b^4c^2+b^2c^4-3c^4a^2+3a^4c^2-3a^2b^2c^2
(4)

and R is the circumradius of the reference triangle.

Its circle function is

 l=-(a^6-b^2a^4-c^2a^4+b^4a^2+c^4a^2-b^2c^2a^2-b^6-c^6+b^2c^4+b^4c^2)/(2bc(a^2-b^2)(a^2-c^2)),
(5)

which is not a Kimberling center.

SecondSteinerCircleNinePointCircle

It passes through Kimberling center X_(114), which is also one of two points in which it intersects the nine-point circle, the other point P having triangle function

 alpha_P=((b^2-c^2)^2(2a^2-b^2-c^2)(a^4-b^4-c^4+b^2c^2))/a
(6)

(P. Moses, pers. comm., Dec. 31, 2004). Furthermore, the line (X_(114),P) (which is the radical line of the second Steiner circle and nine-point circle) is parallel to the Euler line of the reference triangle DeltaABC and passes through X_(30) and X_(2482).


See also

Steiner Circle, Steiner Triangle

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Cite this as:

Weisstein, Eric W. "Second Steiner Circle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SecondSteinerCircle.html

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