- 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.