The Hofstadter ellipses are a family of triangle ellipses introduced by P. Moses in February 2005. The Hofstadter ellipse for parameter is defined by the trilinear equation
(1)
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where
(2)
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(3)
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(4)
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The ellipses and are identical. They are plotted above for , 0.2, ..., 0.5.
The center of the Hofstadter ellipse is given by triangle center function
(5)
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(P. Moses, pers. comm., Feb. 13, 2005), which does not correspond to any Kimberling center.
The Hofstadter ellipse is an inellipse given by
(6)
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and passes through Kimberling centers for , 678, 2310, 2632, 2638, and 2643. It has center
(7)
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corresponding to Kimberling center , which is the crosspoint of the incenter and triangle centroid . It has area
(8)
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where is the area of the reference triangle.
Taking the limit as (or ) gives the circumellipse with trilinear equation
(9)
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This has center with trilinear center function
(10)
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and its fourth intersection with the circumcircle (other than the vertices , , ) given by triangle center function
(11)
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