An inellipse inconic that is an ellipse.
The locus of the centers of the ellipses inscribed in a triangle is the interior of the medial triangle. Newton gave the solution to inscribing an ellipse in a convex quadrilateral (Dörrie 1965, p. 217).
The area of an inellipse with center having areal coordinates 
inscribed in a triangle is
 |
(1)
|
where 
is the area
of the reference triangle (Chakerian 1979,
pp. 143 and 148), which corresponds to an inellipse with center having exact
trilinear coordinates 
having area
 |  |  |
(2)
|
 |  |  |
(3)
|
In terms of the inconic parameters 
, the formula is even simpler,
 |
(4)
|
(E. W. Weisstein, Dec. 4, 2005).
The following table summarizes the areas of some special inellipses.
| inellipse | center | area |
| Brocard inellipse |  |  |
Hofstadter ellipse
with  |  |  |
| incircle |  |  |
| Lemoine inellipse |  |  |
| MacBeath inconic |  |  |
| Mandart inellipse |  | ![(16piDelta^3)/((a+b+c)[2ab+2bc+2ca-(a^2+b^2+c^2)]^(3/2))](/images/equations/Inellipse/Inline23.svg) |
| orthic inconic |  |  |
| Steiner inellipse |  |  |
The centers of the ellipses inscribed in a quadrilateral all lie on the straight line segment joining the midpoints of the polygon diagonals (Chakerian 1979, pp. 136-139).
