section_path: Applications/Geometry/Triangles
section_name: Triangles
paragraph_text: Triangles
  
  Every triangle has a unique Steiner inellipse – an ellipse inside the triangle and tangent to the midpoints of the three sides of the triangle. The foci of a triangle's Steiner inellipse can be found as follows, according to Marden's theorem: Denote the triangle's vertices in the complex plane as a = x + y i, b = x + y i, and c = x + y i. Write the cubic equation (x − a) (x − b) (x − c) = 0 {\displaystyle (x-a)(x-b)(x-c)=0}, take its derivative, and equate the (quadratic) derivative to zero. Marden's theorem says that the solutions of this equation are the complex numbers denoting the locations of the two foci of the Steiner inellipse.