The Bickart points are the foci 
and 
of the Steiner circumellipse.
They have trilinear coordinates 
and 
, where
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
where
 |
(4)
|
and for some appropriate choices of signs. These can be rewritten in fully closed form as
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
where 
and
![Q=sqrt(2(a^2b^2c^2Z^3)-2[b^4c^4(2S^2-a^2S_omega)+c^4a^4(2S^2-b^2S_omega)+a^4b^4(2S^2-c^2S_omega)])](/images/equations/BickartPoints/NumberedEquation2.svg) |
(8)
|
(P. Moses, pers. comm., Mar. 31, 2006), where 
is Conway triangle
notation.
The foci of the Steiner inellipse are given by the same equation, but instead taking 
.
