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Yff Parabola


YffParabola

An inconic with parameters

 x:y:z=a(b-c):b(c-a):c(a-b),
(1)

giving equation

 a^2(b-c)^2alpha^2+b^2(c-a)^2beta^2+c^2(a-b)^2gamma^2-2[ab(b-c)(c-a)alphabeta+ac(a-b)(b-c)alphagamma+bc(a-b)(c-a)betagamma]=0
(2)

(Kimberling 1998, pp. 238-239).

Its focus is Kimberling center X_(101) and its conic section directrix is the line joining the orthocenter H and mittenpunkt M. Its Brianchon point is Kimberling center X_(190), which has equivalent triangle center functions

alpha_(190)=1/(a(b-c))
(3)
alpha_(190)=(bc)/(b-c).
(4)

Its points of tangency with the sides and their extensions form the Yff contact triangle, which is also its polar triangle.

It passes through the points X_(514) and X_(649).

The dual of the Yff parabola is the (nonrectangular) circumhyperbola ABCX_2X_7 having trilinear equation

 ab(a-b)alphabeta+bc(b-c)betagamma+ca(c-a)alphagamma=0
(5)

and center at Kimberling center X_(1086). It passes through Kimberling centers X_i for i=2, 7, 27, 75, 86, 234, 272, 273, 310, 335, 554, 673, 675, 871, 903, 1081, 1088, 1223, 1240, 1246, 1268, 1440, 1659, 2296, 2400, and 2989. It is the isotomic conjugate of the line (X_1,X_2), the isogonal conjugate of the line (X_6,X_(31)), and has the following intersections with various circumconics (P. Moses, pers. comm., Jan. 2005).


See also

Inconic, Parabola, Yff Hyperbola, Yff Contact Triangle

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References

Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.Kimberling, C. "Encyclopedia of Triangle Centers: X(190)=Yff parabolic point." http://faculty.evansville.edu/ck6/encyclopedia/ETC.html#X190.

Referenced on Wolfram|Alpha

Yff Parabola

Cite this as:

Weisstein, Eric W. "Yff Parabola." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/YffParabola.html

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