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Evans Conic


EvansConic

The Evans conic is the conic section passing through the Fermat points X and X^', the inner and outer Napoleon points N and N^', and the isodynamic points S and S^' of a triangle. It has trilinear equation

 a^2alpha^2(-3S^2+S_A^2)(-S^2+3S_A^2)(S_B-S_C)^2+2bcbetagamma(S_A-S_B)(-S_A+S_C)[3a^4S_A^2+(a^2S_A+S_BS_C)(S_B^2-6S_BS_C+SS_C^2)]+[cyclic trilinears],

where S, S_A, S_B, and S_C is Conway triangle notation (P. Moses, pers. comm., Jan. 12, 2005).

The center of the Evans conic has center function

 alpha_(3054)=(12Delta)/a-acotomega,

where Delta is the area of the reference triangle and omega is the Brocard angle (P. Moses, pers. comm., Jan. 12, 2005), which is Kimberling center X_(3054).

The Evans conic passes through Kimberling centers X_i for i=13 (first Fermat point), 14 (second Fermat point), 15 (first isodynamic point), 16 (second isodynamic point), 17 (first Napoleon point), 18 (second Napoleon point), 590, and 615.


See also

Conic Section

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References

Evans, L. S. "A Conic Through Six Triangle Centers." Forum Geom. 2, 89-92, 2002. http://forumgeom.fau.edu/FG2002volume2/FG200211index.html.

Referenced on Wolfram|Alpha

Evans Conic

Cite this as:

Weisstein, Eric W. "Evans Conic." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EvansConic.html

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