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The Complex Cubic and the Steiner Inellipse

Take the vertices of triangle ABC and treat them as complex numbers. They are the zeros of a complex cubic . As you drag point z along the line DE, the image f(z) moves along the image of the line under . As you drag points D and E so the line passes through the triangle, you will see the image curve get looped and kinked. This occurs because the line is passing through or . These are the critical point of the function , the zeros of . Geometrically, these points are the foci of the Steiner Inellipse, the unique inscribed ellipse tangent to the sides of the triangle at the midpoints. The center of this ellipse is the centroid, with is the zero of .