Displaying similar documents to “Some Facts about Trigonometry and Euclidean Geometry”

Circumcenter, Circumcircle and Centroid of a Triangle

Roland Coghetto (2016)

Formalized Mathematics

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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...

Altitude, Orthocenter of a Triangle and Triangulation

Roland Coghetto (2016)

Formalized Mathematics

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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.

Morley’s Trisector Theorem

Roland Coghetto (2015)

Formalized Mathematics

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Morley’s trisector theorem states that “The points of intersection of the adjacent trisectors of the angles of any triangle are the vertices of an equilateral triangle” [10]. There are many proofs of Morley’s trisector theorem [12, 16, 9, 13, 8, 20, 3, 18]. We follow the proof given by A. Letac in [15].

Geometric and combinatorial structure of a class of spherical folding tessellations – I

Catarina P. Avelino, Altino F. Santos (2017)

Czechoslovak Mathematical Journal

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A classification of dihedral folding tessellations of the sphere whose prototiles are a kite and an equilateral or isosceles triangle was obtained in recent four papers by Avelino and Santos (2012, 2013, 2014 and 2015). In this paper we extend this classification, presenting all dihedral folding tessellations of the sphere by kites and scalene triangles in which the shorter side of the kite is equal to the longest side of the triangle. Within two possible cases of adjacency, only one...