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.