The Brocard inellipse is the inconic with parameters
 |
(1)
|
giving the trilinear equation
 |
(2)
|
It has the Brocard points 
and 
of a triangle as its foci and the Brocard
midpoint as its center.
Its Brianchon point is the symmedian point 
of the triangle, and the triangle formed by the points of contact of the inellipse
with the reference triangle is the symmedial
triangle, which is also its polar triangle.
It has semiaxes lengths
 |  |  |
(3)
|
 |  |  |
(4)
|
giving area
 |
(5)
|
where 
is the area of the reference triangle.
It passes through the points 
, 
, 
, 
, and 
(Weisstein, Oct. 16 and Nov. 22, 2004).
This inconic is always an ellipse.
