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Sine-Triple-Angle Circle


Sine-Triple-AngleCircle

Inscribe two triangles DeltaA_1B_1C_1 and DeltaA_2B_2C_2 in a reference triangle DeltaABC such that

A=∠AB_1C_1=∠AC_2B_2
(1)
B=∠BC_1A_1=∠BA_2C_2
(2)
C=∠CA_1B_1=∠CB_2A_2.
(3)

Then the triangles DeltaA_1B_1C_1 and DeltaA_2B_2C_2 are both inscribed in a circle known as the sine-triple-angle circle.

This circle is a central circle having circle function

 l=-(2(c^2+ab-b^2)(b^2+ab-c^2)(c^2+ac-b^2)(b^2+ac-c^2)cosA)/(a^4b^2c^2(1+8cosAcosBcosC)^2).
(4)

The center has triangle center function

 alpha=cos(3A)
(5)

(Kimberling 1998, p. 74), which is Kimberling center X_(49), and the circumradius is

 R_(STA)=R/(|1+8cosAcosBcosC|)
(6)

where R is the circumradius of DeltaABC (Tucker and Neuberg 1887; Thébault 1956; Kimberling 1998, p. 234; typo corrected).

Te sine-triple-angle circle passes through Kimberling centers X_i for i=3043, 3044, 3045, 3046, 3047, and 3048.

The distances satisfy the relationship

 A_1A_2:B_1B_2:C_1C_2=sin(3A):sin(3B):sin(3C).
(7)

(Thébault 1956), which gives the circle its name. The sine-triple-angle circle was originally called the cercle triplicateur by Tucker and Neuberg (1887).

In fact there are infinitely many circles that cut of the side line chords in the same proportions. The centers of these circles lie on the equilateral hyperbola through the in- and excenters and X_(49) (Ehrmann and van Lamoen 2002).

The similitude centers of the nine-point circle and the sine-triple-angle circle are the Kosnita point and the focus of the Kiepert parabola.


See also

Central Circle

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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.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.Kimberling, C. "Encyclopedia of Triangle Centers: X(49)=Center of Sine-Triple-Angle Circle." http://faculty.evansville.edu/ck6/encyclopedia/ETC.html#X49.Thébault, V. "Sine-Triple-Angle-Circle." Mathesis 65, 282-284, 1956.Tucker and Neuberg, J. Mathesis 12, 1887.

Referenced on Wolfram|Alpha

Sine-Triple-Angle Circle

Cite this as:

Weisstein, Eric W. "Sine-Triple-Angle Circle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Sine-Triple-AngleCircle.html

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