Barycentric calculus in Euclidean and hyperbolic geometry by Ungar A.A.

By Ungar A.A.

The note barycentric is derived from the Greek be aware barys (heavy), and refers to heart of gravity. Barycentric calculus is a technique of treating geometry by means of contemplating some degree because the middle of gravity of yes different issues to which weights are ascribed. as a result, specifically, barycentric calculus presents first-class perception into triangle facilities. This special booklet on barycentric calculus in Euclidean and hyperbolic geometry presents an creation to the attention-grabbing and lovely topic of novel triangle facilities in hyperbolic geometry besides analogies they proportion with wide-spread triangle facilities in Euclidean geometry. As such, the publication uncovers awesome unifying notions that Euclidean and hyperbolic triangle facilities proportion. In his prior books the writer followed Cartesian coordinates, trigonometry and vector algebra to be used in hyperbolic geometry that's totally analogous to the typical use of Cartesian coordinates, trigonometry and vector algebra in Euclidean geometry. for that reason, robust instruments which are as a rule on hand in Euclidean geometry turned to be had in hyperbolic geometry in addition, allowing one to discover hyperbolic geometry in novel methods. specifically, this new ebook establishes hyperbolic barycentric coordinates which are used to figure out a variety of hyperbolic triangle facilities simply as Euclidean barycentric coordinates are universal to figure out a number of Euclidean triangle facilities. the search for Euclidean triangle facilities is an previous culture in Euclidean geometry, leading to a repertoire of greater than 3 thousand triangle facilities which are recognized via their barycentric coordinate representations. the purpose of this ebook is to begin an absolutely analogous hunt for hyperbolic triangle facilities that might develop the repertoire of hyperbolic triangle facilities supplied right here.

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63) and the equation sin ∠A1 P3 A3 = sin ∠A2 P3 A3 in Fig. 6, the angle bisector of an angle in a triangle divides the opposite side in the same ratio as the sides adjacent to the angle. Hence, in the notation of Fig. 98) PSfrag replacements May 25, 2010 13:33 WSPC/Book Trim Size for 9in x 6in 30 Barycentric Calculus I a13 A3 γ12 = γa12 = γa12 γ13 = γa13 = γa13 γ23 = γa23 = γa23 ws-book9x6 P2 I a12 = −A1 + A2 , a12 = a12 a13 = −A1 + A3 , a13 = a13 a23 = −A2 + A3 , a23 = a23 3 ∠A1 A2 P2 = ∠A3 A2 P2 a12 A1 ∠A2 A1 P1 = ∠A3 A1 P1 a2 P1 A2 P3 ∠A1 A3 P3 = ∠A2 A3 P3 p1 = −A1 + P1 , p1 = p 1 p2 = −A2 + P2 , p2 = p 2 p3 = −A3 + P3 , p3 = p 3 α1 = ∠A2 A1 A3 , α2 = ∠A1 A2 A3 , α3 = ∠A1 A3 A2 Fig.

7, p. 30. Let P3 be a point on side A1 A2 of triangle A1 A2 A3 in a Euclidean n space Rn such that A3 P3 is an angle bisector of angle ∠A1 A3 A2 , as shown in Fig. 6. 99) below. 91) −m1 a12 m1 (−A2 + A1 ) = p2 = −A2 + P3 = m1 + m 2 m1 + m 2 As indicated in Fig. 95) PSfrag replacements May 25, 2010 13:33 WSPC/Book Trim Size for 9in x 6in Euclidean Barycentric Coordinates ws-book9x6 29 π/2 a12 = −A1 + A2 , a13 = −A1 + A3 , a23 = −A2 + A3 , h = −A3 + P3 , a2 h a13 A3 b 3 a12 A1 a12 = a12 a13 = a13 a23 = a23 h= h A2 P3 p1 = −A1 + P3 , p1 = p 1 p2 = −A2 + P3 , p2 = p 2 α1 = ∠A2 A1 A3 , α2 = ∠A1 A2 A3 , α3 = ∠A1 A3 A2 ∠A1 A3 P3 = ∠A2 A3 P3 Fig.

Let A1 A2 A3 be a triangle in a Euclidean space Rn . Then, in the standard triangle notation, Fig. 14 it is appropriate to present the well-known Heron’s formula [Coxeter (1961)]. 15 (Heron’s Formula). Let A1 A2 A3 be a triangle in a Euclidean space Rn . Then, in the standard triangle notation, Fig. 74), p. 23. 195), p. 63. 126) Triangle Circumcenter The triangle circumcenter is located at the intersection of the perpendicular bisectors of its sides, Fig. 9. Accordingly, it is equidistant from the triangle vertices.

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