An inconic with parameters
 |
(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](/images/equations/YffParabola/NumberedEquation2.svg) |
(2)
|
(Kimberling 1998, pp. 238-239).
Its focus is Kimberling center 
and its conic
section directrix is the line joining the orthocenter 
and mittenpunkt 
. Its Brianchon
point is Kimberling center 
, which has equivalent triangle
center functions
 |  |  |
(3)
|
 |  |  |
(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 
and 
.
The dual of the Yff parabola is the (nonrectangular) circumhyperbola 
having trilinear equation
 |
(5)
|
and center at Kimberling center 
. It passes through Kimberling centers 
for 
, 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 
, the isogonal
conjugate of the line 
,
and has the following intersections with various circumconics (P. Moses, pers.
comm., Jan. 2005).
