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Formalized Mathematics

(a computer assisted approach)

Editor-in-Chief: Matuszewski, Roman

4 Issues per year


SCImago Journal Rank (SJR) 2015: 0.134
Source Normalized Impact per Paper (SNIP) 2015: 0.686
Impact per Publication (IPP) 2015: 0.296

Open Access
Online
ISSN
1898-9934
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In This Section
Volume 24, Issue 1 (Mar 2016)

Issues

Altitude, Orthocenter of a Triangle and Triangulation

Roland Coghetto
  • Rue de la Brasserie 5 7100 La Louvière, Belgium
Published Online: 2016-08-31 | DOI: https://doi.org/10.1515/forma-2016-0003

Summary

We introduce the altitudes of a triangle (the cevians perpendicular to the opposite sides). Using the generalized Ceva’s Theorem, we prove the existence and uniqueness of the orthocenter of a triangle [7]. Finally, we formalize in Mizar [1] some formulas [2] to calculate distance using triangulation.

MSC: 51M04; 03B35

Keywords: Euclidean geometry; trigonometry; altitude; orthocenter; triangulation; distance

MML : identifier: EUCLID13; version: 8.1.04 5.36.1267

References

  • [1] Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pąk, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261-279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-8 17. [Crossref]

  • [2] R. Campbell. La trigonométrie. Que sais-je? Presses universitaires de France, 1956.

  • [3] Wenpai Chang, Yatsuka Nakamura, and Piotr Rudnicki. Inner products and angles of complex numbers. Formalized Mathematics, 11(3):275-280, 2003.

  • [4] Roland Coghetto. Some facts about trigonometry and Euclidean geometry. Formalized Mathematics, 22(4):313-319, 2014. doi:10.2478/forma-2014-0031. [Crossref]

  • [5] Roland Coghetto. Morley’s trisector theorem. Formalized Mathematics, 23(2):75-79, 2015. doi:10.1515/forma-2015-0007. [Crossref]

  • [6] Roland Coghetto. Circumcenter, circumcircle and centroid of a triangle. Formalized Mathematics, 24(1):19-29, 2016. doi:10.1515/forma-2016-0002. [Crossref]

  • [7] H.S.M. Coxeter and S.L. Greitzer. Geometry Revisited. The Mathematical Association of America (Inc.), 1967.

  • [8] Akihiro Kubo. Lines on planes in n-dimensional Euclidean spaces. Formalized Mathematics, 13(3):389-397, 2005.

  • [9] Akihiro Kubo. Lines in n-dimensional Euclidean spaces. Formalized Mathematics, 11(4): 371-376, 2003.

  • [10] Akihiro Kubo and Yatsuka Nakamura. Angle and triangle in Euclidean topological space. Formalized Mathematics, 11(3):281-287, 2003.

  • [11] Marco Riccardi. Heron’s formula and Ptolemy’s theorem. Formalized Mathematics, 16 (2):97-101, 2008. doi:10.2478/v10037-008-0014-2. [Crossref]

  • [12] Boris A. Shminke. Routh’s, Menelaus’ and generalized Ceva’s theorems. Formalized Mathematics, 20(2):157-159, 2012. doi:10.2478/v10037-012-0018-9. [Crossref]

  • [13] Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3):445-449, 1990.

  • [14] Yuguang Yang and Yasunari Shidama. Trigonometric functions and existence of circle ratio. Formalized Mathematics, 7(2):255-263, 1998.

About the article

Received: 2015-12-30

Published Online: 2016-08-31

Published in Print: 2016-03-01



Citation Information: Formalized Mathematics, ISSN (Online) 1898-9934, DOI: https://doi.org/10.1515/forma-2016-0003. Export Citation

© by Roland Coghetto. This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License. (CC BY-SA 3.0)

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