Yff Hyperbola


The Yff hyperbola is the hyperbola given parametrically by


The trilinear equation is complicated expression with coefficients up to degree 10 in the side lengths.

This hyperbola has vertices at the triangle centroid G and orthocenter H, a focus at the circumcenter O, and a directrix given by the line passing through the nine-point center N and perpendicular to the Euler line (Yff 1987; Kimberling 1998, p. 244).

Its center is therefore the midpoint of GH, which is Kimberling center X_(381).

Its transverse axis length a^' and focal distance c are


where R is the circumradius of the reference triangle, so the eccentricity of the hyperbola is


giving the remarkable result that this hyperbola has the same eccentricity in every triangle except for the equilateral triangle (which has no Euler line and no Yff hyperbola; P. Yff, pers. comm.).

The only Kimberling centers through which is passes are X_i=2 (triangle centroid G) and 4 (orthocenter H).

See also


Explore with Wolfram|Alpha


Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.Yff, P. "On the beta-Lines and beta-Circles of a Triangle." Ann. New York Acad. Sci. 500, 561-569, 1987.

Referenced on Wolfram|Alpha

Yff Hyperbola

Cite this as:

Weisstein, Eric W. "Yff Hyperbola." From MathWorld--A Wolfram Web Resource.

Subject classifications