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Displaying similar documents to “Rotation indices related to Poncelet’s closure theorem”

Rotation indices related to Poncelet’s closure theorem

Waldemar Cieślak, Horst Martini, Witold Mozgawa (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let CRCr denote an annulus formed by two non-concentric circles CR, Cr in the Euclidean plane. We prove that if Poncelet’s closure theorem holds for k-gons circuminscribed to CRCr, then there exist circles inside this annulus which satisfy Poncelet’s closure theorem together with Cr, with n- gons for any n > k.

Some Facts about Trigonometry and Euclidean Geometry

Roland Coghetto (2014)

Formalized Mathematics

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We calculate the values of the trigonometric functions for angles: [XXX] , by [16]. After defining some trigonometric identities, we demonstrate conventional trigonometric formulas in the triangle, and the geometric property, by [14], of the triangle inscribed in a semicircle, by the proposition 3.31 in [15]. Then we define the diameter of the circumscribed circle of a triangle using the definition of the area of a triangle and prove some identities of a triangle [9]. We conclude by...

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].