# Steiner Inellipse

- Author:
- Steve Phelps

The

**Steiner inellipse**,**midpoint inellipse**, or**midpoint ellipse**of a triangle is the unique ellipse inscribed in the triangle and tangent to the sides at their midpoints. The Steiner inellipse has the maximum area of any inellipse.## A Connection to Complex Numbers

Consider the vertices of triangle

*ABC*to be complex numbers. Furthermore, consider them to the be the roots of the complex polynomial f(z) = (z – A)(z – B)(z – C). The critical points of this polynomial - where f'(z) = 0 - are the foci of the ellipse. Furthermore, the centroid (as a complex number) is the solution to f''(z) = 0. You can find out more by reading the article below.