TOPICS
Search

Second Steiner Circle

DOWNLOAD Mathematica NotebookDownload Wolfram Notebook
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

Explore with Wolfram|Alpha

Cite this as:

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

Subject classifications