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.
@article{bwmeta1.element.doi-10_1515_forma-2016-0002,
author = {Roland Coghetto},
title = {Circumcenter, Circumcircle and Centroid of a Triangle},
journal = {Formalized Mathematics},
volume = {24},
year = {2016},
pages = {17-26},
zbl = {1343.51008},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1515_forma-2016-0002}
}
Roland Coghetto. Circumcenter, Circumcircle and Centroid of a Triangle. Formalized Mathematics, Tome 24 (2016) pp. 17-26. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1515_forma-2016-0002/
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[3] H.S.M. Coxeter and S.L. Greitzer. Geometry Revisited. The Mathematical Association of America (Inc.), 1967. | Zbl 0166.16402
[4] Robin Hartshorne. Geometry: Euclid and beyond. Springer, 2000.
[5] Akihiro Kubo. Lines on planes in n-dimensional Euclidean spaces. Formalized Mathematics, 13(3):389-397, 2005.
[6] Marco Riccardi. Heron’s formula and Ptolemy’s theorem. Formalized Mathematics, 16(2): 97-101, 2008. doi:10.2478/v10037-008-0014-2.