Unique ellipse tangent to all 3 midpoints of a given triangle's sides
The Steiner Inellipse. According to Marden's theorem, given the triangle with vertices (1, 7), (7, 5), (3, 1), the foci of the inellipse are (3, 5) and (13/3, 11/3), since
In geometry, the Steiner inellipse,[1]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 inellipse. 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[2] to Jakob Steiner, and a proof of its uniqueness is given by Dan Kalman.[3]
The Steiner inellipse contrasts with the Steiner circumellipse, also called simply the Steiner ellipse, which is the unique ellipse that passes through the vertices of a given triangle and whose center is the triangle's centroid.[4]
^Weisstein, E. "Steiner Inellipse" — From MathWorld, A Wolfram Web Resource, http://mathworld.wolfram.com/SteinerInellipse.html.
^H. Dörrie, 100 Great Problems of Elementary Mathematics, Their History and Solution (trans. D. Antin), Dover, New York, 1965, problem 98.
^Kalman, Dan (2008), "An elementary proof of Marden's theorem" (PDF), American Mathematical Monthly, 115 (4): 330–338, JSTOR 27642475, MR 2398412, archived from the original (PDF) on 2012-08-26.
^Weisstein, Eric W. "Steiner Circumellipse". MathWorld.
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