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Steiner Inellipse


SteinerInellipse

The Steiner inellipse, also called the midpoint ellipse (Chakerian 1979), is an inellipse with inconic parameters

 x:y:z=a:b:c
(1)

giving equation

 a^2alpha^2+b^2beta^2+c^2gamma^2-2abalphabeta-2acalphagamma-2bcbetagamma=0.
(2)

It has the triangle centroid G as both its center and Brianchon point.

It is tangent to the sides of the triangle at their midpoints, so the triangle formed by its contact points with the reference triangle is the medial triangle, which is also its polar triangle.

This conic is always an ellipse.

The Steiner inellipse has the maximum area of any inellipse (Chakerian 1979). Under an affine transformation, the Steiner inellipse can be transformed into the incircle of an equilateral triangle.

It passes through Kimberling centers X_i for i=115 (the center of the Kiepert hyperbola), 1015, 1084, 1086, 1146, 2454, 2455, and 2482 (the antipode of 115).

It is the image of the Steiner circumellipse in the homothecy with homothetic center G and similitude ratio 1/2. Therefore, the Steiner inellipse has semiaxes lengths

a=1/6sqrt(a^2+b^2+c^2+2Z)
(3)
b=1/6sqrt(a^2+b^2+c^2-2Z),
(4)

where

 Z=sqrt(a^4+b^4+c^4-a^2b^2-b^2c^2-c^2a^2),
(5)

and area

 A=pi/(3sqrt(3))Delta,
(6)

where Delta is the area of the reference triangle (P. Moses, pers. comm., Dec. 31, 2004).

The Steiner inellipse is the Steiner circumellipse of the medial triangle.


See also

Affine Transformation, Brianchon Point, Midpoint, Inconic, Inellipse, Medial Triangle, Steiner Circumellipse

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References

Chakerian, G. D. "A Distorted View of Geometry." Ch. 7 in Mathematical Plums (Ed. R. Honsberger). Washington, DC: Math. Assoc. Amer., pp. 135-136 and 145-146, 1979.Coxeter, H. S. M. and Pedoe, D. Central Similarities. Minneapolis, MN: University of Minnesota College Geometry Project, 1971. 10 min. Distributed by International Film Bureau, Inc. (Chicago, IL). http://mkat.iwf.de/index.asp?Signatur=W+1429.Pedoe, D. "Thinking Geometrically." Amer. Math. Monthly 77, 711-721, 1970.

Referenced on Wolfram|Alpha

Steiner Inellipse

Cite this as:

Weisstein, Eric W. "Steiner Inellipse." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SteinerInellipse.html

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