1. bookVolume 24 (2016): Issue 1 (March 2016)
Journal Details
License
Format
Journal
eISSN
1898-9934
First Published
09 Jun 2008
Publication timeframe
4 times per year
Languages
English
Open Access

Circumcenter, Circumcircle and Centroid of a Triangle

Published Online: 31 Aug 2016
Volume & Issue: Volume 24 (2016) - Issue 1 (March 2016)
Page range: 17 - 26
Received: 30 Dec 2015
Journal Details
License
Format
Journal
eISSN
1898-9934
First Published
09 Jun 2008
Publication timeframe
4 times per year
Languages
English
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.

Keywords

MSC

MML

[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 17.10.1007/978-3-319-20615-8Search in Google Scholar

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

[3] H.S.M. Coxeter and S.L. Greitzer. Geometry Revisited. The Mathematical Association of America (Inc.), 1967.10.5948/UPO9780883859346Search in Google Scholar

[4] Robin Hartshorne. Geometry: Euclid and beyond. Springer, 2000.10.1007/978-0-387-22676-7Search in Google Scholar

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

[6] Marco Riccardi. Heron’s formula and Ptolemy’s theorem. Formalized Mathematics, 16(2): 97-101, 2008. doi:10.2478/v10037-008-0014-2.10.2478/v10037-008-0014-2Search in Google Scholar

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