A new characterization for isometries by triangles.

Li, Baokui; Wang, Yuefei

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Displaying similar documents to “A new characterization for isometries by triangles.”

Properties of triangulations obtained by the longest-edge bisection

Francisco Perdomo, Ángel Plaza (2014)

Open Mathematics

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The Longest-Edge (LE) bisection of a triangle is obtained by joining the midpoint of its longest edge with the opposite vertex. Here two properties of the longest-edge bisection scheme for triangles are proved. For any triangle, the number of distinct triangles (up to similarity) generated by longest-edge bisection is finite. In addition, if LE-bisection is iteratively applied to an initial triangle, then minimum angle of the resulting triangles is greater or equal than a half of the...

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...

On the existence of triangle with given angle and opposite angle bisectors length.

Bukor, József (2008)

Annales Mathematicae et Informaticae

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The envelope of the Wallace-Simpson lines of a triangle. A simple proof of the Steiner problem on the deltoid.

Miguel de Guzmán (2001)

RACSAM

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A simple proof is presented of a famous, and difficult, theorem by Jakob Steiner. By means of a straightforward transformation of the triangle, the proof of the theorem is reduced to the case of the equilateral triangle. Several relations of the Steiner deltoid with the Feuerbach circle and the Morley triangle appear then as obvious.

Dividing the Sides of a Triangle in Proportional Parts

Paulus Gerdes (2003)

Visual Mathematics

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Hyperbolas and orthologic triangles.

Čerin, Z. (1997)

Mathematica Pannonica

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Cevians as sides of triangles.

Čerin, Zvonko (2000)

Mathematica Pannonica

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Cutting circles and the Morley theorem.

Stammler, Ludwig (1997)

Beiträge zur Algebra und Geometrie

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Divisible tilings in the hyperbolic plane.

Broughton, S.Allen, Haney, Dawn M., McKeough, Lori T., Smith Mayfield, Brandy (2000)

The New York Journal of Mathematics [electronic only]

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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.

Pascal like triangles and Sierpinski like gaskets

Tomohide Hashiba, Yuta Nakagawa, Toshiyuki Yamauchi, Hiroshi Matsui, Satoshi Hashiba, Daisuke Minematsu, Munetoshi Sakaguchi, Ryohei Miyadera (2007)

Visual Mathematics

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On the geometric relation of twin-points of the triangle. (Zur Verwandtschaft der Zwillingspunkte beim Dreieck.)

Schaal, Hermann (1993)

Beiträge zur Algebra und Geometrie

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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].

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...