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Translation of the Kimberling’s Glossary into barycentrics (v48-dvipdfm) June 13, 2012 Acknowledgement. This document began its life as a private copy of the Glossary accompanying the Encyclopedia of Clark Kimberling. This Glossary - as its name suggests - is organized alpha- betically. As a –never satisfied– newcomer, I would have preferred a progression from the easiest to the hardest topics and I reordered this document in my own way. It is unclear whether this new ordering will be useful to someone else ! In any case a detailed index is provided. Second point, the Glossary is written using trilinear coordinates. From an advanced point of view, these coordinates are neither better nor worse than the barycentric coordinates. Nevertheless, having some practice of the barycentrics, and none of the trilinears, I undertook to translate everything, from one system to another. In any case, this was a formative exercise, and this also puts the focus on the covariance/contravariance properties that were subsequently systematized. Drawings are the third point. Everybody knows –or should know– that geometry is not possible at all without drawings. Having no intention to pay royalties for using rulers and compasses, I turned to an open source software (kseg) in order to produce my own drawings. Thereafter, I have used Geogebra, together with pstricks. What a battle, but no progress without practice ! Subsequently, other elements have been incorporated from other sources, including materials about cubics, from the Gibert web site. Finally, original elements were also added. As it will appear at first sight, "pldx" is addicted to a tradition that requires a precisely specified universal space for each object to live in. A second massive addiction of the author is computer algebra. Having at your fingertips a tool that gives the right answer to each and every expansion or factorization, and never lost the small paper sheet where the computation of the week was summarized is really great. Moreover, being constrained to explain everything to a computer helps to specify all the required details. For example, the "equivalence up to a proportionality factor" doesn’t apply in the same way to a matrix whether this matrix describes a collineation, a triangle, a trigon or a set of incidence relations. In this document, "beautiful geometrical proofs" are avoided as most as possible, since they are the most error prone. A safe proof of "the triangle of contacts of the inscribed circle and the triangle of the mid-arcs on the circumcircle admit the similicenter of the two circles as perspector (and therefore as center of similitude)" is : ency(persp(matcev(pX(7)),matucev(pX(1)))) 7! 56 where the crucial point is ency, i.e. a safe implementation of the Kimberling’s search key method to explore the database. To summarize, the present document is rather a "derived work", where the elements presented are not intended to be genuine, apart perhaps from the way to assemble the pieces and cook them all together. Index =well known circles, 108 cevadiv, 41, 150 =well known conics, 85 cevadivision, 151 =well known cubics, 166 cevamul, 40, 150 =well known lines, 24 cevapoint, 41 =well known triangles, 33 cevian collineation, 150 cevian conics, 86 A cevian conjugacies, 151 affix (complex), 137 cevian lines, 32 affix (Morley), 139 cevian nest, 38 affix, Morley, 140 cevian triangle, 32 alephdivision, 151 circle, 105 Al-Kashi, 46, 50, 106 circle (list of well known circles), 108 angle between lines, 54 circlekit, 106 anticevian triangle, 34 circum-anticevian, 71 anticomplement, 31, 150 circumcevian, 71 anticomplementary conjugate, 195 circumcircle, 85, 109 anticomplementary triangle, 31, 33 circumconic, 82 antigonal conjugacy, 163 circumconic (circumhyperbola), 95 antimedial triangle, 31 circumconic, asymptote, 95 antiorthic axis, 24 circumhyperbola, 95 Apollonius circle, 112 cK(F,U), 176 Arbelos, 117 cocevian triangle, 34 area (complex formula), 137 collinear, 18 area (Heron), 50 collineation, 147 area (matrix), 50 collineation algorithm, 147 Collings transform, 199 B backward matrix, 140 complement, 31, 150 backward substitutions, 142 complementary conjugate, 195 Barycentric division, 21 complex affix, 137 barycentric multiplication, 21, 149 concurrent, 19 barycentrics, 18, 21 conic, 77 barycentrics (standard), 19 Conic-cross-ratio, 156 base points (pencil of circles), 126, 161 conico-pivotal isocubic, 176 Bevan circle, 112 conjugate circle, 111 Bevan line, 24 cosinus of projection, 58 bicentric points, 27 cross conjugacy, 39 Brocard 3-6 circle, 115 cross ratio of four points, 32 Brocard ellipse, 92 crossdiff, 196 Brocard line, 24 crossdifference, 23 Brocard points, 56 crossdiv, 39, 150 Brocard second circle, 115 crossmul, 38, 150 crosspoint, 39 C cross-ratio, 155 center of a circle, 106 crosssum, 196 center of a conic, 80 crosssumbar, 196 central line, 27 crosssumdiv, 196 central point, 27 crosstriangle, 36 central triangle, 29 cubic, 165 ceva conjugacy, 41 cubic (list of well known cubics), 166 3 4 INDEX cubic (triangular), 168 hyperbola (inhyperbola), 96 cubic cK(F,U), 176 cubic nK(P,U,k), 175 I cubic nK0(P,U), 175 incentral triangle, 33 cubic pK(P,U), 166 incircle, 85, 86, 110 cubics PKA, 168 Incircle transform, 110 cycle, 122 inconic, 83 cyclocevian conjugate, 119 inconic (inhyperbola), 96 cyclopedal conjugate, 60 infinity line, 24 inhyperbola, 96 D intouch triangle, 33 dalethdivision, 153 inverse in a circle, 108 Daneels (first) perspector, 42 involutive collineation, 148 Darboux cubic, 171 isoceles triangles, 117 DC point, 73 Isoconjugacy, 161 de Longchamps axis, 24 isogonal conjugacy, 60 desmic mate, 73 isogonal conjugacy (Morley), 162 distance point-line, 56 isoptic pencil, 126 distance point-point, 51 isotomic conjugacy, 34 dual of a conic, 80 isotomic pencil, 125 Dual triangle, 73 duality, 22 J Jerabek RH, 85 E eigencenter, 154 K eigentransform, 174 K219, 96 embedded vector, 50 K170 cubic, 167, 174, 177 Euler line, 24 K004 Darboux, 171 excentral triangle, 33 K155 EAC2, 173 Exceter point, 71 K007 Lucas, 171 extouch triangle, 33 K003 McKay, 169 K001 Neuberg, 170 F K010 Simson, 177 focus, 88 K002 Thomson, 171 folium, 77 K-ellipse, 92 forward matrix, 140 Kiepert parabola, 85, 143 forward substitutions, 140 Kiepert RH, 85, 117 fourth harmonic, 32, 155 K-matrix, 50 Fuhrmann 4-8 circle, 116 kseg, 20 Fuhrmann triangle, 33, 192 L G Lemoine axis, 24 geogebra, 20 Lemoine first circle, 113 Gergonne axis, 24 Lemoine inconic, 85 Gibert-Simson transform, 177 Lemoine second circle, 114 gimeldivision, 152 limit points (pencil of circles), 125, 161 Gt mapping, 101 line, 18 gudulic point, 88 line (list of well known lines), 24 line at infinity, 19, 141 H Longchamps circle, 112 harmonic division, 32 Lubin parametrization, 140 helethdivision, 153 Lucas cubic, 171 Hirst inverse, 196 hirstpoint, 196 M homoltriangle, 36 MacBeath circumconic, 86 homothecy, 58 MacBeath inconic, 86 horizon, 122 major center, 27 hyperbola, 94 matrix K, 50 hyperbola (circumhyperbola), 95 matrix M, 54, 141 June 13, 2012 14:18 published under the GNU Free Documentation License INDEX 5 matrix OrtO, 53, 141 polynomial center, 27 matrix Pyth, 51, 141 power, 51 matrix W, 49, 141 power formula, 51, 105 McKay cubic K003, 169 projector (orthogonal), 57 medial triangle, 31, 33 psi-Kimberling, 151 Menelaus theorem, 35 Pythagoras theorem, 51 Miguel theorem, 35 Pyth-matrix, 51 mimosa, 152 Morley affix, 139, 140 R Morley space, 138 radical trace, 107 multiplier of an homography, 158 radius of a circle, 106, 125 rational center, 28 N reciprocal conjugacy, 161 Nagel line, 24 rectangular hyperbola, 95 Neuberg circles, 56 reflection, 31 Neuberg K001 cubic, 170 representative, 122 nine points circle, 86 RH, 95 nine-points circle, 111 Rigby point, 65 nK(P,U,k), 175 rotation, 149 nK0(P,U), 175 RS, 65 non pivotal isocubic, 165 S O Saragossa points, 73 orthic axis, 24, 62 search key, Kimberling, 19 orthic triangle, 33 search key, Morley, 20, 142 orthoassociate, 67 Second Daneels perspector, 43 orthocentroidal 2-4 circle, 116 shortest cubic, 170 orthocorrespondent, 67 similitude, 148 orthodir, 54, 141 similitude (centers of), 107 orthogonal projector, 57, 149 Simpson line, 61 orthogonal reflection, 58, 149 Simson cubic, 177 orthojoin, 64 Simson-Moses point, 66 orthologic (triangles), 67 sine-triple-angle circle, 114 Orthopoint, 53 SM, 66 orthopole, 62 Soddy (but not so) circles, 131 OrtO-matrix, 53 Soddy circles, 130 Soddy line, 24 P Sondat, 181 parabola, 81, 93 Spieker circle, 112 pedal triangle, 59 sqrtdiv, 22, 159 Pelle à Tarte, 117 sqrtmul, 22 pencil, 123 squarediv, 42 perspectivity, 36 squaremul, 42 perspectivity with ABC, 37 SR, 65 perspector, 36, 37 Star triangle, 192 perspector of a conic, 81 star triangle, 33 perspectrix, 36, 37 Steinbart transform, 71 pivot of a cubic, 165 Steiner circumellipse, 43, 85 pivotal conic, 176 Steiner inellipse, 85, 90 pivotal umbilical pK, 169 Steiner line, 61 PK transform, 167 symbolic substitution, 29 pK(P,U), 166 symmetrical functions, 140 Plucker formulas, 79 symmetry (orthogonal), 58 point, 18 polar wrt a conic, 80 T polar wrt a curve, 77 tangent, 54 polars-of-points triangle, 153 tangent of two lines, 54, 141 pole wrt a conic, 80 tangent to curve, 77 poles-of-lines triangle, 153 tangential triangle, 33 —– pldx : Translation of the Kimberling’s Glossary into barycentrics —– 6 INDEX Tarry point, 57 TCC-perspector, 71 Tg mapping, 101 Thomson K002 cubic, 171 transcendental center, 27 translation, 148 transpose, 23 triangle, 18 triangle (list of well known triangles), 33 triangle (may be degenerate), 18 triangle center, 27 trigone, 19 trilinear pole (deprecated), 23 Trilinears, 21 tripolar, 23 tripolar centroid, 36 tripolar line, 34 tripole, 23 type-crossing, 20 type-keeping, 20 U umbilics, 122 unary cofactor triangle, 154 V vandermonde, 140 Vertex associate, 74 vertex triangle, 36 vertex trigon, 36 visible point, 139 W wedge operator, 22 W-matrix, 49 Y Yff parabola, 85 Z Z(U) cubic, 168 Z+cubic, 176 zosma, 153 June 13, 2012 14:18 published under the GNU Free Documentation License List of Figures 1.1 Point P and line A”B”C” are the tripolars of each other.
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  • Coincidence of Centers for Scalene Triangles

    Coincidence of Centers for Scalene Triangles

    Forum Geometricorum b Volume 7 (2007) 137–146. bbb FORUM GEOM ISSN 1534-1178 Coincidence of Centers for Scalene Triangles Sadi Abu-Saymeh and Mowaffaq Hajja Abstract.Acenter function is a function Z that assigns to every triangle T in a Euclidean plane E a point Z(T ) in E in a manner that is symmetric and that respects isometries and dilations. A family F of center functions is said to be complete if for every scalene triangle ABC and every point P in its plane, there is Z∈F such that Z(ABC)=P . It is said to be separating if no two center functions in F coincide for any scalene triangle. In this note, we give simple examples of complete separating families of continuous triangle center functions. Regarding the impression that no two different center functions can coincide on a scalene triangle, we show that for every center function Z and every scalene triangle T , there is another center function Z, of a simple type, such that Z(T )=Z(T ). 1. Introduction Exercise 1 of [33, p. 37] states that if any two of the four classical centers coin- cide for a triangle, then it is equilateral. This can be seen by proving each of the 6 substatements involved, as is done for example in [26, pp. 78–79], and it also follows from more interesting considerations as described in Remark 5 below. The statement is still true if one adds the Gergonne, the Nagel, and the Fermat-Torricelli centers to the list. Here again, one proves each of the relevant 21 substatements; see [15], where variants of these 21 substatements are proved.