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 
).
-Lines and 
-Circles of a Triangle." Ann. New York Acad. Sci. 500,
561-569, 1987.