Triangle inequality (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Triangle inequality" in English language version.

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8books.google.com

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4archive.org

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3wolframalpha.com

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2doi.org

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1wolfram.com

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1clarku.edu

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1ams.org

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1worldcat.org

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1arxiv.org

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ams.org (Global: 451^st place; English: 277^th place)

Izumi, Shuzo (2016). "Sufficiency of simplex inequalities". Proceedings of the American Mathematical Society. 144 (3): 1299–1307. doi:10.1090/proc12756. ISSN 0002-9939.

archive.org (Global: 6^th place; English: 6^th place)

Arlan Ramsay; Robert D. Richtmyer (1995). Introduction to hyperbolic geometry. Springer. p. 17. ISBN 0-387-94339-0.
Claude Irwin Palmer (1919). Practical mathematics for home study: being the essentials of arithmetic, geometry, algebra and trigonometry. McGraw-Hill. p. 422.
A proof not requiring separate cases is as follows: Any number is always less than or equal to its own absolute value, so $-|u|\leq u\leq |u|$ and $-|v|\leq v\leq |v|.$ Adding these inequalities together, $-{\bigl (}|u|+|v|{\bigr )}\leq u+v\leq |u|+|v|,$ the sum has the form $-a\leq b\leq a$ for $a=|u|+|v|$ and $b=u+v.$ But this always implies $|b|\leq a$ , or, expanded, $|u+v|\leq |u|+|v|.$
James Stewart (2008). Essential Calculus. Thomson Brooks/Cole. p. A10. ISBN 978-0-495-10860-3.
John Stillwell (2005). The four pillars of geometry. Springer. p. 80. ISBN 0-387-25530-3.

arxiv.org (Global: 69^th place; English: 59^th place)

Schubert, Erich (2021). A Triangle Inequality for Cosine Similarity. International Conference on Similarity Search and Applications. Dortmund: Springer. arXiv:2107.04071. doi:10.1007/978-3-030-89657-7_3.

books.google.com (Global: 3^rd place; English: 3^rd place)

Mohamed A. Khamsi; William A. Kirk (2001). "§1.4 The triangle inequality in $R n$ ". An introduction to metric spaces and fixed point theory. Wiley-IEEE. ISBN 0-471-41825-0.
Oliver Brock; Jeff Trinkle; Fabio Ramos (2009). Robotics: Science and Systems IV. MIT Press. p. 195. ISBN 978-0-262-51309-8.
Harold R. Jacobs (2003). Geometry: seeing, doing, understanding (3rd ed.). Macmillan. p. 201. ISBN 0-7167-4361-2.
Alexander Zawaira; Gavin Hitchcock (2009). "Lemma 1: In a right-angled triangle the hypotenuse is greater than either of the other two sides". A primer for mathematics competitions. Oxford University Press. ISBN 978-0-19-953988-8.
John Stillwell (1997). Numbers and Geometry. Springer. ISBN 978-0-387-98289-2. p. 95.
Rainer Kress (1988). "§3.1: Normed spaces". Numerical analysis. Springer. p. 26. ISBN 0-387-98408-9.
Karen Saxe (2002). Beginning functional analysis. Springer. p. 61. ISBN 0-387-95224-1.
Anonymous (1854). "Exercise I. to proposition XIX". The popular educator; fourth volume. Ludgate Hill, London: John Cassell. p. 196.

clarku.edu (Global: 6,602^nd place; English: 6,109^th place)

mathcs.clarku.edu

David E. Joyce (1997). "Euclid's Elements, Book I, Proposition 20". Euclid's Elements. Dept. Math and Computer Science, Clark University. Retrieved 2010-06-25.

doi.org (Global: 2^nd place; English: 2^nd place)

Izumi, Shuzo (2016). "Sufficiency of simplex inequalities". Proceedings of the American Mathematical Society. 144 (3): 1299–1307. doi:10.1090/proc12756. ISSN 0002-9939.
Schubert, Erich (2021). A Triangle Inequality for Cosine Similarity. International Conference on Similarity Search and Applications. Dortmund: Springer. arXiv:2107.04071. doi:10.1007/978-3-030-89657-7_3.

wolfram.com (Global: 513^th place; English: 537^th place)

mathworld.wolfram.com

Wolfram MathWorld – http://mathworld.wolfram.com/TriangleInequality.html

wolframalpha.com (Global: 4,660^th place; English: 5,295^th place)

Wolfram|Alpha. "input: solve 0<a<2a+3d, 0<a+d<2a+2d, 0<a+2d<2a+d,". Wolfram Research. Retrieved 2010-09-07.
Wolfram|Alpha. "input: solve 0<a<ar+ar², 0<ar<a+ar², 0<ar²<a+ar". Wolfram Research. Retrieved 2010-09-07.
Wolfram|Alpha. "input: solve 0<a<ar+ar²+ar³, 0<ar³<a+ar+ar²". Wolfram Research. Retrieved 2012-07-29.

worldcat.org (Global: 5^th place; English: 5^th place)

search.worldcat.org

Izumi, Shuzo (2016). "Sufficiency of simplex inequalities". Proceedings of the American Mathematical Society. 144 (3): 1299–1307. doi:10.1090/proc12756. ISSN 0002-9939.

Triangle inequality (English Wikipedia)

ams.org (Global: 451st place; English: 277th place)

archive.org (Global: 6th place; English: 6th place)

arxiv.org (Global: 69th place; English: 59th place)

books.google.com (Global: 3rd place; English: 3rd place)

clarku.edu (Global: 6,602nd place; English: 6,109th place)