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Formalized Mathematics

(a computer assisted approach)

Editor-in-Chief: Matuszewski, Roman

4 Issues per year


SCImago Journal Rank (SJR) 2016: 0.207
Source Normalized Impact per Paper (SNIP) 2016: 0.315

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

Issues

Circumcenter, Circumcircle and Centroid of a Triangle

Roland Coghetto
Published Online: 2016-08-31 | DOI: https://doi.org/10.1515/forma-2016-0002

Summary

We introduce, using the Mizar system [1], some basic concepts of Euclidean geometry: the half length and the midpoint of a segment, the perpendicular bisector of a segment, the medians (the cevians that join the vertices of a triangle to the midpoints of the opposite sides) of a triangle.

We prove the existence and uniqueness of the circumcenter of a triangle (the intersection of the three perpendicular bisectors of the sides of the triangle). The extended law of sines and the formula of the radius of the Morley’s trisector triangle are formalized [3].

Using the generalized Ceva’s Theorem, we prove the existence and uniqueness of the centroid (the common point of the medians [4]) of a triangle.

MSC: 51M04; 03B35

Keywords: Euclidean geometry; perpendicular bisector; circumcenter; circumcircle; centroid; extended law of sines

MML: identifier: EUCLID12; 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.CrossrefGoogle Scholar

  • [2] Czesław Byliński. Introduction to real linear topological spaces. Formalized Mathematics, 13(1):99-107, 2005.Google Scholar

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

  • [4] Robin Hartshorne. Geometry: Euclid and beyond. Springer, 2000.Google Scholar

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

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

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-2016-0002.

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

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