Journal of Geometry

, Volume 107, Issue 2, pp 305–316 | Cite as

Remarks on orthocenters, Pappus’ theorem and Butterfly theorems

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Abstract

We present a generalization of the notion of the orthocenter of a triangle and of Pappus’ theorem. Both subjects were discussed with Pickert in the last year of his life. Furthermore we add a projective Butterfly theorem which covers all known affine cases.

Keywords

Orthocenter Pappus’ theorem Butterfly theorem 

Mathematics Subject Classification

Primary 51M04 Secondary 50A30 

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References

  1. 1.
    Bankoff, L.: The Metamorphosis of the Butterfly problem. Mathematics magazine 60, 195–210 (1987). doi: 10.2307/2689339
  2. 2.
    Coxeter, H.S.M.: Projective Geometry, 2nd edn. University of Toronto Press, (1974)Google Scholar
  3. 3.
    Coxeter, H.S.M., Greitzer, S.L.: Geometry Revisited. Fifth printing, The Mathematical Association of America, (1967). http://www.aproged.pt/biblioteca/geometryrevisited_coxetergreitzer
  4. 4.
    Fritsch, R., Pickert, G.: Schwerpunkte von Vierecken. Die Wurzel 48, 35–41, 74–81, 90–95 (2014); Englisch version: Crux Mathematicorum 39, 178–184, 266–272, 362–367 (2013); see also http://www.math.lmu.de/~fritsch/Viereckschwerpunkt
  5. 5.
    Geupel, O., Pickert, G.: Solution of problem 3848. Crux Mathematicorum 40, 226–227, (2014)Google Scholar
  6. 6.
    Horner, W.G., et al., Answered Question 1029. The Gentleman’s Diary, 39–40 (1815). http://babel.hathitrust.org/cgi/pt?id=njp.32101076451903;view=1up;seq=5
  7. 7.
    Mackay, J.S.: Geometrical notes I. Proc. Edinb. Math. Soc. 3, 38–40 (1884/85). doi: 10.1017/S0013091500037251
  8. 8.
    Möbius, A.F.: Verallgemeinerung des Pascalschen Theorems, das in einen Kegelschnitt beschriebene Sechseck betreffend. J. Reine Angew. Math. 36, 216–220 (1848). http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002146258
  9. 9.
    O’Connor, J.J., Robertson, E.F.: The Edinburgh Mathematical Society 1883–1933. http://www-history.mcs.st-and.ac.uk/HistTopics/EMS_history.html
  10. 10.
    O’Connor, J.J., Robertson, E.F.: William George Horner. http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Horner.html
  11. 11.
    Scurr, T.: Question 1029. The Gentleman’s Diary, 47 (1814). http://babel.hathitrust.org/cgi/pt?id=njp.32101076451903;view=1up;seq=5
  12. 12.
    Veblen, O., Young, J.W.: Projective Geometry. Ginn and Company, (1910). https://archive.org/details/projectivegeomet01vebluoft
  13. 13.
    Volenec, V.: The butterfly theorem for conics. Math. Commun. 7, 35–38, (2002). http://hrcak.srce.hr/file/1512

Copyright information

© Springer International Publishing 2015

Authors and Affiliations

  1. 1.Mathematisches InstitutLudwig-Maximilians-Universität MünchenMunichGermany

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