Morgan's Theorem
Morgan's theorem is a generalization of Marion's theorem found by Ryan Morgan, a sophomore at Patapsco High School in Baltimore (Morgan 1994). If the sides of a triangle are partitioned into n equal segments for n an odd integer and each division point is connected to the opposite vertex, a central hexagon is formed. Morgan's theorem states that the area of this hexagon relative to the original triangle is
(Morgan 1994, Watanabe et al. 1996). For n=1, 3, 5, ..., this gives one over the centered nonagonal numbers 1, 10, 28, 55, 91, 136, 190, 253, 325, 406, ... (OEIS A060544).
See also
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References
Johanson, D. "Re: Marion's Theorem." geom.pre-college discussion group, Jan 7, 1995. https://web.archive.org/web/20040823092021id_/http://mathforum.org/epigone/geom.pre-college/111/3emro37ドルep@newsbf02.news.aol.com.Morgan, R. "No Restriction Needed." Math. Teacher 87, 726 and 743, 1994.Sloane, N. J. A. Sequence A060544 in "The On-Line Encyclopedia of Integer Sequences."Walter, M. "Re: Morgan's Theorem?" geometry-forum discussion group, Feb 3, 1995. https://web.archive.org/web/20030327123509/http://mathforum.org/epigone/geometry-forum/27.Watanabe, T.; Hanson, R.; and Nowosielski, F. D. "Morgan's Theorem." Math. Teacher 89, 420-423, 1996.Referenced on Wolfram|Alpha
Morgan's TheoremCite this as:
Weisstein, Eric W. "Morgan's Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MorgansTheorem.html