The New Proof of Ptolemy’s Theorem & Nine Point Circle Theorem

Dasari Naga Vijay Krishna

doi:doi:10.11648/j.mcs.20160104.14

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The New Proof of Ptolemy’s Theorem & Nine Point Circle Theorem

Dasari Naga Vijay Krishna

Published in Mathematics and Computer Science (Volume 1, Issue 4)

Received: 31 August 2016 Accepted: 18 October 2016 Published: 14 December 2016

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Abstract

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.

Published in	Mathematics and Computer Science (Volume 1, Issue 4)
DOI	10.11648/j.mcs.20160104.14
Page(s)	93-100
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2016. Published by Science Publishing Group

Keywords

Ptolemy’s Theorem, Circumcenter, Cyclic Quadrilateral, Nine Point Circle Theorem, Pedals Triangle, Medial Triangle

References

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[2]	Clark Kimberling, Twenty – one points on the nine-point circle, The Mathematical Gazette, Vol. 92, No. 523 (March2008), pp.29-38.
[3]	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.
[4]	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.
[5]	Erwin Just Norman Schaumerger, A Vector Approach to Ptolemy's Theorem, Mathematics Magzine, Vol.77, NO.5, 2004.
[6]	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 .
[7]	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.
[8]	J. L Coolidge, A Historically Interesting Formula for the Area of a Cyclic Quadrilateral, Amer. Math. Monthly, 46(1939), pp. 345–347.
[9]	Michael de Villiers, A Generalization of the Nine-point circle and Euler line, Pythagoras, 62, Dec05, pp.31-35.
[10]	Michael de Villiers, The nine-point conic: a rediscovery and proof by computer, International Journal of Mathematical Education in Science and Technology, vol37, 2006.
[11]	Mehmet Efe Akengin, Zeyd Yusuf Koroglu, Yigit Yargi, Three Natural Homoteties of The Nine-Point Circle, Forum Geometricorum, Volume13 (2013) 209–218.
[12]	Martin Josefsson, Properties of Equidiagonal Quadrilaterals, Forum Geometricorum, Volume 14 (2014) 129–144.
[13]	O. Shisha, On Ptolemy’s Theorem, International Journal of Mathematics and Mathematical Sciences, 14.2 (1991) p.410.
[14]	Sidney H. Kung, Proof without Words: The Law of Cosines via Ptolemy's Theorem, Mathematics Magazine, April, 1992.
[15]	Shay Gueron, Two Applications of the Generalized Ptolemy Theorem, The Mathematical Association of America, Monthly109, 2002.
[16]	S. Shirali, On the generalized Ptolemy theorem, Crux Math. 22 (1989) 49-53.
[17]	http://skepticbutjewish.blogspot.in/2010/11/generalizing-ptolemys-theorem.html.
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[19]	http://www.vedicbooks.net/geometry-in-ancient-and-medieval-india-p-637.html.
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Dasari Naga Vijay Krishna. (2016). The New Proof of Ptolemy’s Theorem & Nine Point Circle Theorem. Mathematics and Computer Science, 1(4), 93-100. https://doi.org/10.11648/j.mcs.20160104.14

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ACS Style

Dasari Naga Vijay Krishna. The New Proof of Ptolemy’s Theorem & Nine Point Circle Theorem. Math. Comput. Sci. 2016, 1(4), 93-100. doi: 10.11648/j.mcs.20160104.14

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AMA Style

Dasari Naga Vijay Krishna. The New Proof of Ptolemy’s Theorem & Nine Point Circle Theorem. Math Comput Sci. 2016;1(4):93-100. doi: 10.11648/j.mcs.20160104.14

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@article{10.11648/j.mcs.20160104.14,
  author = {Dasari Naga Vijay Krishna},
  title = {The New Proof of Ptolemy’s Theorem & Nine Point Circle Theorem},
  journal = {Mathematics and Computer Science},
  volume = {1},
  number = {4},
  pages = {93-100},
  doi = {10.11648/j.mcs.20160104.14},
  url = {https://doi.org/10.11648/j.mcs.20160104.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mcs.20160104.14},
  abstract = {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.},
 year = {2016}
}

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TY  - JOUR
T1  - The New Proof of Ptolemy’s Theorem & Nine Point Circle Theorem
AU  - Dasari Naga Vijay Krishna
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PY  - 2016
N1  - https://doi.org/10.11648/j.mcs.20160104.14
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T2  - Mathematics and Computer Science
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SP  - 93
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PB  - Science Publishing Group
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AB  - 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.
VL  - 1
IS  - 4
ER  -

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Author Information

Dasari Naga Vijay Krishna

Department of Mathematics, Narayana Educational Instutions, Bengalore, India

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Dasari Naga Vijay Krishna. (2016). The New Proof of Ptolemy’s Theorem & Nine Point Circle Theorem. Mathematics and Computer Science, 1(4), 93-100. https://doi.org/10.11648/j.mcs.20160104.14

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ACS Style

Dasari Naga Vijay Krishna. The New Proof of Ptolemy’s Theorem & Nine Point Circle Theorem. Math. Comput. Sci. 2016, 1(4), 93-100. doi: 10.11648/j.mcs.20160104.14

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AMA Style

Dasari Naga Vijay Krishna. The New Proof of Ptolemy’s Theorem & Nine Point Circle Theorem. Math Comput Sci. 2016;1(4):93-100. doi: 10.11648/j.mcs.20160104.14

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@article{10.11648/j.mcs.20160104.14,
  author = {Dasari Naga Vijay Krishna},
  title = {The New Proof of Ptolemy’s Theorem & Nine Point Circle Theorem},
  journal = {Mathematics and Computer Science},
  volume = {1},
  number = {4},
  pages = {93-100},
  doi = {10.11648/j.mcs.20160104.14},
  url = {https://doi.org/10.11648/j.mcs.20160104.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mcs.20160104.14},
  abstract = {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.},
 year = {2016}
}

Copy | Download

TY  - JOUR
T1  - The New Proof of Ptolemy’s Theorem & Nine Point Circle Theorem
AU  - Dasari Naga Vijay Krishna
Y1  - 2016/12/14
PY  - 2016
N1  - https://doi.org/10.11648/j.mcs.20160104.14
DO  - 10.11648/j.mcs.20160104.14
T2  - Mathematics and Computer Science
JF  - Mathematics and Computer Science
JO  - Mathematics and Computer Science
SP  - 93
EP  - 100
PB  - Science Publishing Group
SN  - 2575-6028
UR  - https://doi.org/10.11648/j.mcs.20160104.14
AB  - 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.
VL  - 1
IS  - 4
ER  -

Copy | Download