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Stammler Circles


StammlerCircles

The Stammler circles are the three circles (apart from the circumcircle), that intercept the sidelines of a reference triangle DeltaABC in chords of lengths equal to the corresponding side lengths a, b, and c. The centers of these circles form an equilateral triangle DeltaO_AO_BO_C known as the Stammler triangle.

The radical lines of pairs of the Stammler circles are the Simson lines of the vertices of the circumtangential triangle.

The trilinear coordinates of the center of the A-Stammler circle are

 cosA-2cos[1/3(B-C)]:cosB+2cos[1/3(B+2C)] 
 :cosC+2cos[1/3(2B+C)]
(1)

and the squares of the radii R_A^2, R_B^2, R_C^2 are given by the roots of the cubic equation

 x^3-R^2[9x^2-3(a^2+b^2+c^2)x+(a^2b^2+b^2c^2+c^2a^2)]=0,
(2)

where R is the circumradius of the reference triangle (Ehrmann and van Lamoen 2002).

Explicitly, the radii of the Stammler circles are

R_A=sqrt(1+8cos[1/3(B-C)]cos[1/3(B+2C)]cos[1/3(2B+C)])R
(3)
R_B=sqrt(1+8cos[1/3(C-A)]cos[1/3(C+2A)]cos[1/3(2C+A)])R
(4)
R_C=sqrt(1+8cos[1/3(A-B)]cos[1/3(A+2B)]cos[1/3(2A+B)])R,
(5)

where R is again the circumradius of the reference triangle (Ehrmann and van Lamoen 2002).

When R_A, R_B, and R_C are the radii of the Stammler circles and R the circumradius, then the following equations hold:

R_A^2+R_B^2+R_C^2=9R^2
(6)
1/(R_A^2)+1/(R_B^2)+1/(R_C^2)=(3(a^2+b^2+c^2))/(a^2b^2+a^2c^2+b^2c^2)
(7)
R_AR_BR_C=Rsqrt(a^2b^2+a^2c^2+b^2c^2)
(8)

(Ehrmann and van Lamoen 2002).


See also

Proportionally-Cutting Circle, Stammler Circle, Stammler Circles Radical Circle, Stammler Hyperbola, Stammler Triangle

This entry contributed by Floor van Lamoen

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References

Ehrmann, J.-P. and van Lamoen, F. M. "The Stammler Circles." Forum Geom. 2, 151-161, 2002. http://forumgeom.fau.edu/FG2002volume2/FG200219index.html.Stammler, L. "Proportionalschnittkreise, ihre Mittenhyperbel und ein Pendant zum Satz von Morley." Elem. Math. 47, 148-158, 1992.Stammler, L. "Cutting Circles and the Morley Theorem." Beitr. Alg. Geom. 38, 91-93, 1997. http://www.emis.de/journals/BAG/vol.38/no.1/7.html.

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Stammler Circles

Cite this as:

van Lamoen, Floor. "Stammler Circles." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/StammlerCircles.html

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