153
Views
0
CrossRef citations to date
0
Altmetric
Articles

The Ptolemy-Euler Theorem via Reflection

&
Pages 208-214 | Received 17 Mar 2022, Accepted 31 Aug 2022, Published online: 15 Mar 2024

References

  • Kung SH. Proof without words: the law of cosines via Ptolemy’s theorem. Math Mag. 1992;65(2):103. DOI: 10.1080/0025570X.1992.11995990.
  • Heo NG. Proof without words: Pythagorean theorem via Ptolemy’s theorem. Math Mag. 2017;90(3):187. DOI: 10.4169/math.mag.90.3.187.
  • Apostol TM. Ptolemy’s inequality and the chordal metric. Math Mag. 1967;40(5):233–235. DOI: 10.2307/2688275.
  • Cohen MR, Drabkin IE. 1948. A source book in Greek science. New York: McGraw-Hill.
  • Hahn LS. Complex numbers and geometry. Providence: Mathematical Association of America; 1994.
  • Gauss CF. Disquisitiones arithmeticae. Clarke AA, translator. New York: Spring-Verlag; 1986.
  • Euclid. The thirteen books of Euclid’s elements. Vol.2 (Books III-IX). Heath TL, translator. Cambridge: Cambridge University Press, 1908.
  • Just E, Schaumberger N. A vector approach to Ptolemy’s theorem. Math Mag. 2004;77(5):386–388. DOI: 10.2307/3219205.
  • Hardy GH. A course of pure mathematics. Cambridge: Cambridge University Press; 1908.
  • Johnson EH, editor. Problems. Math Mag. 2001;74(3):239–246. DOI: 10.1080/0025570X.2001.11953071.
  • Gueron S. Two applications of the generalized Ptolemy theorem. Amer Math Monthly. 2002;109(4):362–370. DOI: 10.1080/00029890.2002.11920897.
  • Chamberland M, Zeilberger D. A short proof of McDougall’s circle theorem. Amer Math Monthly. 2014;121(3):263–265. DOI: 10.4169/amer.math.monthly.121.03.263.
  • Valentine JE. 1970. An analogue of Ptolemy’s theorem in spherical geometry. Amer Math Monthly. 77 (1):47–51. DOI: 10.1080/00029890.1970.11992415.
  • Schoenberg IJ. On metric arcs of vanishing Menger curvature. Ann Math. 1940;41(4):715–726. DOI: 10.2307/1968849.
  • Klamkin MS, Meir A. Ptolemy’s inequality, chordal metric, multiplicative metric. Pac J Math. 1982;101(2):389–392. DOI: 10.2140/pjm.1982.101.389.
  • Dieudonné J. Éléments D’analyse, Fondements de L’analyse Moderne (Tome I). New York: Chez Academic Press; 1979.
  • Wells D. The Penguin dictionary of curious and interesting geometry. London: Penguin Books; 1991.
  • Johnson RA. Advanced Euclidean geometry. Modern geometry: an elementary treatise on the geometry of the triangle and the circle. New York: Dover; 1960.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.