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
).