TOPICS

# Yff Hyperbola

The Yff hyperbola is the hyperbola given parametrically by

 (1)

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

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

Its center is therefore the midpoint of , which is Kimberling center .

Its transverse axis length and focal distance are

 (2) (3)

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

 (4)

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 (triangle centroid ) and 4 (orthocenter ).

Hyperbola

## Explore with Wolfram|Alpha

More things to try:

## References

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

Yff Hyperbola

## Cite this as:

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