The Steiner inellipse, also called the midpoint ellipse (Chakerian 1979), is an inellipse with inconic parameters
 |
(1)
|
giving equation
 |
(2)
|
It has the triangle centroid 
as both its center and Brianchon
point.
It is tangent to the sides of the triangle at their midpoints, so the triangle formed by its contact points with the reference triangle is the medial triangle, which is also its polar triangle.
This conic is always an ellipse.
The Steiner inellipse has the maximum area of any inellipse (Chakerian 1979). Under an affine transformation, the Steiner inellipse can be transformed into the incircle of an equilateral triangle.
It passes through Kimberling centers 
for 
(the center of the Kiepert
hyperbola), 1015, 1084, 1086, 1146, 2454, 2455, and 2482 (the antipode of 115).
It is the image of the Steiner circumellipse in the homothecy with homothetic
center 
and similitude ratio 
. Therefore, the Steiner inellipse has semiaxes lengths
 |  |  |
(3)
|
 |  |  |
(4)
|
where
 |
(5)
|
and area
 |
(6)
|
where 
is the area of the reference triangle (P. Moses,
pers. comm., Dec. 31, 2004).
The Steiner inellipse is the Steiner circumellipse of the medial triangle.
