Steiner inellipse

In geometry, 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. It is an example of an inconic. By comparison the inscribed circle and Mandart inellipse of a triangle are other inconics that are tangent to the sides, but not at the midpoints unless the triangle is equilateral. The Steiner inellipse is attributed by Dörrie to Jakob Steiner, and a proof of its uniqueness is given by Kalman.

The Steiner inellipse contrasts with the Steiner circumellipse, also called simply the Steiner ellipse, which is the unique ellipse that touches a given triangle at its vertices and whose center is the triangle's centroid.

The equation of the Steiner inellipse in trilinear coordinates for a triangle with side lengths a, b, c is

The center of a triangle's Steiner inellipse is the triangle's centroid — the intersection of the triangle's medians. The Steiner inellipse is the only inellipse whose center is at the triangle's centroid.

The Steiner inellipse of a triangle has the largest area of any inellipse of that triangle; as the largest inscribed ellipse, it is the John ellipsoid of the triangle. Its area is ${\displaystyle {\tfrac {\pi }{3{\sqrt {3}}}}}$ times the area of the triangle. Thus its area is one-fourth that of the Steiner circumellipse.

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