Steiner inellipse

[

The Steiner Inellipse. According to Marden's theorem, given the trianglewith vertices (1,7), (7,5) and (3,1), the fociof inellipse are (3,5) and (13/3,11/3), since"Dx"(1 + 7"i" − "x")(7 + 5"i" − "x")(3 + "I" − "x") = (13/3 + 11/3"i" − "x")(3 + 5"i" − "x").]

In geometry, the Steiner inellipse of a triangle, named after Jakob Steiner, is the unique ellipse inscribed in the triangle and tangent to the sides at their midpoints. By comparison, the inscribed circle of a triangle is tangent to the sides, but not necessarily at the midpoints.

According to Marden's theorem, if the three vertices of the triangle are the complex zeros of a cubic polynomial, then the foci of the Steiner inellipse are the zeros of the derivative of the polynomial.