Volume 1, Issue 4, November 2016, Page: 93-100
The New Proof of Ptolemy’s Theorem & Nine Point Circle Theorem
Dasari Naga Vijay Krishna, Department of Mathematics, Narayana Educational Instutions, Bengalore, India
Received: Aug. 31, 2016;       Accepted: Oct. 18, 2016;       Published: Dec. 14, 2016
DOI: 10.11648/j.mcs.20160104.14      View  4900      Downloads  283
The main purpose of the paper is to present a new proof of the two celebrated theorems: one is “Ptolemy's Theorem” which explains the relation between the sides and diagonals of a cyclic quadrilateral and another is “Nine Point Circle Theorem” which states that in any arbitrary triangle the three midpoints of the sides, the three feet of altitudes, the three midpoints of line segments formed by joining the vertices and Orthocenter, total nine points are concyclic. Our new proof is based on a metric relation of circumcenter.
Ptolemy’s Theorem, Circumcenter, Cyclic Quadrilateral, Nine Point Circle Theorem, Pedals Triangle, Medial Triangle
To cite this article
Dasari Naga Vijay Krishna, The New Proof of Ptolemy’s Theorem & Nine Point Circle Theorem, Mathematics and Computer Science. Vol. 1, No. 4, 2016, pp. 93-100. doi: 10.11648/j.mcs.20160104.14
Copyright © 2016 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Claudi Alsina and Roger B. Nelsen, On the Diagonals of a Cyclic Quadrilateral, Forum Geometricorum, Volume 7 (2007)147–149.
Clark Kimberling, Twenty – one points on the nine-point circle, The Mathematical Gazette, Vol. 92, No. 523 (March2008), pp.29-38.
Dasari Naga Vijay Krishna, Distance Between the Circumcenter and Any Point in the Plane of the Triangle, Geo Gebra International Journal of Romania (GGIJRO),volume-5, No. 2, 2016 art 92, pp 139-148.
Dasari Naga Vijay Krishna, Yet another proof of Feuerbach’s Theorem, Global Journal of Science Frontier Research: F, Mathematics and Decision Science, volume-16, issue-4, version-1.0, 2016, p9-15.
Erwin Just Norman Schaumerger, A Vector Approach to Ptolemy's Theorem, Mathematics Magzine, Vol.77, NO.5, 2004.
G. W. Indika Shameera Amarasinghe, A Concise Elementary Proof For The Ptolemy’s Theorem, Global Journal of Advanced Research on Classical and Modern Geometries, Vol. 2, Issue1, 2013, pp. 20-25 .
J. E. Valentine, An Analogue of Ptolemy's Theorem in Spherical Geometry The American Mathematical Monthly, Vol. 77, No.1 (Jan.,1970), pp. 47-51.
J. L Coolidge, A Historically Interesting Formula for the Area of a Cyclic Quadrilateral, Amer. Math. Monthly, 46(1939), pp. 345–347.
Michael de Villiers, A Generalization of the Nine-point circle and Euler line, Pythagoras, 62, Dec05, pp.31-35.
Michael de Villiers, The nine-point conic: a rediscovery and proof by computer, International Journal of Mathematical Education in Science and Technology, vol37, 2006.
Mehmet Efe Akengin, Zeyd Yusuf Koroglu, Yigit Yargi, Three Natural Homoteties of The Nine-Point Circle, Forum Geometricorum, Volume13 (2013) 209–218.
Martin Josefsson, Properties of Equidiagonal Quadrilaterals, Forum Geometricorum, Volume 14 (2014) 129–144.
O. Shisha, On Ptolemy’s Theorem, International Journal of Mathematics and Mathematical Sciences, 14.2 (1991) p.410.
Sidney H. Kung, Proof without Words: The Law of Cosines via Ptolemy's Theorem, Mathematics Magazine, April, 1992.
Shay Gueron, Two Applications of the Generalized Ptolemy Theorem, The Mathematical Association of America, Monthly109, 2002.
S. Shirali, On the generalized Ptolemy theorem, Crux Math. 22 (1989) 49-53.
Browse journals by subject