Altitude, Orthocenter of a Triangle and Triangulation

• Autorzy: Roland Coghetto
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

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

• Wydawca: De Gruyter Open
• Czasopismo: Formalized Mathematics
• Rocznik: 2016
• Tom: 24
• Numer: 1
• Strony: 27-36

Daty

wydano

2016-03-01

otrzymano

2015-12-30

online

2016-08-19

Twórcy

autor

Roland Coghetto

• Rue de la Brasserie 5 7100 La Louvière, Belgium

Bibliografia

• [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.
• [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.
• [5] Roland Coghetto. Morley’s trisector theorem. Formalized Mathematics, 23(2):75-79, 2015. doi:10.1515/forma-2015-0007.
• [6] Roland Coghetto. Circumcenter, circumcircle and centroid of a triangle. Formalized Mathematics, 24(1):19-29, 2016. doi:10.1515/forma-2016-0002.
• [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.
• [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.
• [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.

Identyfikatory

DOI

10.1515/forma-2016-0003