Altitude, Orthocenter of a Triangle and Triangulation

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.Coghetto Roland - Rue de la Brasserie 5 7100 La Louvière, BelgiumGrzegorz 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.R. Campbell. La trigonométrie. Que sais-je? Presses universitaires de France, 1956.Wenpai Chang, Yatsuka Nakamura, and Piotr Rudnicki. Inner products and angles of complex numbers. Formalized Mathematics, 11(3):275-280, 2003.Roland Coghetto. Some facts about trigonometry and Euclidean geometry. Formalized Mathematics, 22(4):313-319, 2014. doi:10.2478/forma-2014-0031.Roland Coghetto. Morley’s trisector theorem. Formalized Mathematics, 23(2):75-79, 2015. doi:10.1515/forma-2015-0007.Roland Coghetto. Circumcenter, circumcircle and centroid of a triangle. Formalized Mathematics, 24(1):19-29, 2016. doi:10.1515/forma-2016-0002.H.S.M. Coxeter and S.L. Greitzer. Geometry Revisited. The Mathematical Association of America (Inc.), 1967.Akihiro Kubo. Lines on planes in n-dimensional Euclidean spaces. Formalized Mathematics, 13(3):389-397, 2005.Akihiro Kubo. Lines in n-dimensional Euclidean spaces. Formalized Mathematics, 11(4): 371-376, 2003.Akihiro Kubo and Yatsuka Nakamura. Angle and triangle in Euclidean topological space. Formalized Mathematics, 11(3):281-287, 2003.Marco Riccardi. Heron’s formula and Ptolemy’s theorem. Formalized Mathematics, 16 (2):97-101, 2008. doi:10.2478/v10037-008-0014-2.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.Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3):445-449, 1990.Yuguang Yang and Yasunari Shidama. Trigonometric functions and existence of circle ratio. Formalized Mathematics, 7(2):255-263, 1998