Sava Grozdev and Deko Dekov, the Lester Circle Is Orthogonal to the Brocard Circle Of

Total Page：16

File Type：pdf, Size：1020Kb

Sava Grozdev and Deko Dekov, the Lester Circle Is Orthogonal to the Brocard Circle Of

Journal of Computer-Generated Mathematics Problem 8 The Lester Circle is orthogonal to the Brocard Circle of the Fourth Brocard Triangle Sava Grozdev and Deko Dekov Submitted on May 1, 2014 Publication Date: July 10, 2014 At the present time, there are seven notable circles known to be orthogonal to the Lester circle. See [1]. The below problem introduces a new notable circle which is orthogonal to the Lester circle. Prove the following problem, produced by the computer program “Discoverer”: Problem 8. Prove that the Lester Circle is orthogonal to the Brocard Circle of the Fourth Brocard Triangle. The reader may find the definitions in [2-5]. Please submit the solution of the problem for publication in this journal to the editor of this journal: [email protected] See the figure: Journal of Computer-Generated Mathematics 2014 Problem no 4 Page 1 of 2 In the figure: c – Lester circle, DEF – Fourth Brocard Triangle, c1 - Brocard Circle of Triangle DEF. Circle c1 is orthogonal to the Lester circle. References 1. Dekov, D., Computer-Generated Mathematics: Seven Circles orthogonal to the Lester Circle, Didactical Modeling, 2008, http://www.math.bas.bg/omi/DidMod/Articles/D%5B1%5D.Dekov_Lester_Circle.pdf 2. Sava Grozdev and Deko Dekov, Computer-Generated Encyclopedia of Euclidean Geometry, 2014, available at the Web: http://www.ddekov.eu/e2/ 3. Eric W. Weisstein, MathWorld - A Wolfram Web Resource, http://mathworld.wolfram.com/ 4. Quim Castellsaguer, The Triangles Web, http://www.xtec.es/~qcastell/ttw/ttweng/portada.html 5. P. Yiu, The Circles of Lester, Evans, Parry, and Their Generalizations, Forum Geometricorum, 10 175–209. Sava Grozdev, Sofia, Bulgaria, [email protected] Deko Dekov, Stara Zagora, Bulgaria, [email protected] Journal of Computer-Generated Mathematics 2014 Problem no 4 Page 2 of 2 .

Copy Link

Recommended publications

Circles in Areals
Circles in areals 23 December 2015 This paper has been accepted for publication in the Mathematical Gazette, and copyright is assigned to the Mathematical Association (UK). This preprint may be read or downloaded, and used for any non-profit making educational or research purpose. Introduction August Möbiusintroduced the system of barycentric or areal co-ordinates in 1827[1, 2]. The idea is that one may attach weights to points, and that a system of weights determines a centre of mass. Given a triangle ABC, one obtains a co-ordinate system for the plane by placing weights x; y and z at the vertices (with x + y + z = 1) to describe the point which is the centre of mass. The vertices have co-ordinates A = (1; 0; 0), B = (0; 1; 0) and C = (0; 0; 1). By scaling so that triangle ABC has area [ABC] = 1, we can take the co-ordinates of P to be ([P BC]; [PCA]; [P AB]), provided that we take area to be signed. Our convention is that anticlockwise triangles have positive area. Points inside the triangle have strictly positive co-ordinates, and points outside the triangle must have at least one negative co-ordinate (we are allowed negative masses). The equation of a line looks like the equation of a plane in Cartesian co-ordinates, but note that the equation lx + ly + lz = 0 (with l 6= 0) is not satisfied by any point (x; y; z) of the Euclidean plane. We use such equations to describe the line at infinity when doing projective geometry.
[Show full text]
On a Construction of Hagge
Forum Geometricorum b Volume 7 (2007) 231–247. b b FORUM GEOM ISSN 1534-1178 On a Construction of Hagge Christopher J. Bradley and Geoff C. Smith Abstract. In 1907 Hagge constructed a circle associated with each cevian point P of triangle ABC. If P is on the circumcircle this circle degenerates to a straight line through the orthocenter which is parallel to the Wallace-Simson line of P . We give a new proof of Hagge’s result by a method based on reflections. We introduce an axis associated with the construction, and (via an areal anal- ysis) a conic which generalizes the nine-point circle. The precise locus of the orthocenter in a Brocard porism is identified by using Hagge’s theorem as a tool. Other natural loci associated with Hagge’s construction are discussed. 1. Introduction One hundred years ago, Karl Hagge wrote an article in Zeitschrift fur¨ Mathema- tischen und Naturwissenschaftliche Unterricht entitled (in loose translation) “The Fuhrmann and Brocard circles as special cases of a general circle construction” [5]. In this paper he managed to find an elegant extension of the Wallace-Simson theorem when the generating point is not on the circumcircle. Instead of creating a line, one makes a circle through seven important points. In 2 we give a new proof of the correctness of Hagge’s construction, extend and appl§ y the idea in various ways. As a tribute to Hagge’s beautiful insight, we present this work as a cente- nary celebration. Note that the name Hagge is also associated with other circles [6], but here we refer only to the construction just described.
[Show full text]
The Radii by R, Ra, Rb,Rc
The Orthopole, by R. GOORMAGHTIGH,La Louviero, Belgium. Introduction. 1. In this little work are presented the more important resear- ches on one of the newest chapters in the modern Geometry of the triangle, and whose origin is, the following theorem due to. Professor J. Neuberg: Let ƒ¿, ƒÀ, y be the projections of A, B, C on a straight line m; the perpendiculars from a on BC, ƒÀ on CA, ƒÁ on ABare concurrent(1). The point of concurrence. M is called the orthopole of m for the triangle ABC. The following general conventions will now be observed : the sides of the triangle of reference ABC will be denoted by a, b, c; the centre and radius of the circumcircle by 0 and R; the orthocentre by H; the centroid by G; the in- and ex-centresby I, Ia 1b, Ic and the radii by r, ra, rb,rc;the mid-points of BC, CA, AB by Am, Bm, Cm;the feet of the perpendiculars AH, BH, OH from A, B, C on BO, CA, AB by An,Bn, Cn; the contact points of the incircle with BC, CA, AB by D, E, F; the corresponding points for the circles Ia, Ib, Ic, by Da, Ea,Fa, Dc, Eb, F0, Dc, Ec, Fc; the area of ABC by 4; the centre of the nine-point circle by 0g; the Lemoine . point by K; the Brocard points by theGergonne and Nagel points by and N(2); further 2s=a+b+c. Two points whose joins to A, B, C are isogonal conjugate in the angles A, B, C are counterpoints.
[Show full text]
A History of Elementary Mathematics, with Hints on Methods of Teaching
;-NRLF I 1 UNIVERSITY OF CALIFORNIA PEFARTMENT OF CIVIL ENGINEERING BERKELEY, CALIFORNIA Engineering Library A HISTORY OF ELEMENTARY MATHEMATICS THE MACMILLAN COMPANY NEW YORK BOSTON CHICAGO DALLAS ATLANTA SAN FRANCISCO MACMILLAN & CO., LIMITED LONDON BOMBAY CALCUTTA MELBOURNE THE MACMILLAN CO. OF CANADA, LTD. TORONTO A HISTORY OF ELEMENTARY MATHEMATICS WITH HINTS ON METHODS OF TEACHING BY FLORIAN CAJORI, PH.D. PROFESSOR OF MATHEMATICS IN COLORADO COLLEGE REVISED AND ENLARGED EDITION THE MACMILLAN COMPANY LONDON : MACMILLAN & CO., LTD. 1917 All rights reserved Engineering Library COPYRIGHT, 1896 AND 1917, BY THE MACMILLAN COMPANY. Set up and electrotyped September, 1896. Reprinted August, 1897; March, 1905; October, 1907; August, 1910; February, 1914. Revised and enlarged edition, February, 1917. o ^ PREFACE TO THE FIRST EDITION "THE education of the child must accord both in mode and arrangement with the education of mankind as consid- ered in other the of historically ; or, words, genesis knowledge in the individual must follow the same course as the genesis of knowledge in the race. To M. Comte we believe society owes the enunciation of this doctrine a doctrine which we may accept without committing ourselves to his theory of 1 the genesis of knowledge, either in its causes or its order." If this principle, held also by Pestalozzi and Froebel, be correct, then it would seem as if the knowledge of the history of a science must be an effectual aid in teaching that science. Be this doctrine true or false, certainly the experience of many instructors establishes the importance 2 of mathematical history in teaching. With the hope of being of some assistance to my fellow-teachers, I have pre- pared this book and have interlined my narrative with occasional remarks and suggestions on methods of teaching.
[Show full text]
The Symmedian Point
Page 1 of 36 Computer-Generated Mathematics: The Symmedian Point Deko Dekov Abstract. We illustrate the use of the computer program "Machine for Questions and Answers" (The Machine) for discovering of new theorems in Euclidean Geometry. The paper contains more than 100 new theorems about the Symmedian Point, discovered by the Machine. Keywords: computer-generated mathematics, Euclidean geometry "Within ten years a digital computer will discover and prove an important mathematical theorem." (Simon and Newell, 1958). This is the famous prediction by Simon and Newell [1]. Now is 2008, 50 years later. The first computer program able easily to discover new deep mathematical theorems - The Machine for Questions and Answers (The Machine) [2,3] has been created by the author of this article, in 2006, that is, 48 years after the prediction. The Machine has discovered a few thousands new mathematical theorems, that is, more than 90% of the new mathematical computer-generated theorems since the prediction by Simon and Newell. In 2006, the Machine has produced the first computer-generated encyclopedia [2]. Given an object (point, triangle, circle, line, etc.), the Machine produces theorems related to the properties of the object. The theorems produced by the Machine are either known theorems, or possible new theorems. A possible new theorem means that the theorem is either known theorem, but the source is not available for the author of the Machine, or the theorem is a new theorem. I expect that approximately 75 to 90% of the possible new theorems are new theorems. Although the Machine works completely independent from the human thinking, the theorems produced are surprisingly similar to the theorems produced by the people.
[Show full text]
1 1. the Brocard Circle and Its Isogonal Conjugate Locus The
1. The Brocard circle and its isogonal conjugate locus The equation of the Brocard circle is b2c2x2 + c2a2y2 + a2b2z2 – a4yz – b4zx – c4xy = 0 (1.1) It is known, of course that this has OK as a diameter, where O is the circumcentre of triangle ABC and K is the symmedian point. Also, the two Brocard points W(1/b2, 1/c2, 1/a2) and W'(1/c2, 1/a2, 1/b2) lie on the circle, with the chord WW' perpendicular to OK. The isogonal conjugate locus of the Brocard circle ha equation a2y2z2 + b2z2x2 + c2x2y2 – xyz(a2x + b2y + c2z). (1.2) This contains the orthocentre H, the centroid G and again, the pair of Brocard points W and W'. 2. The six defining circles The six defining circles of the Brocard points are: BCW: – b2x2 + (c2 – b2)xy + a2yz = 0, (2.1) CAW: b2zx – c2y2 + (a2 – c2)yz = 0, (2.2) ABW: – a2z2 + c2xy + (b2 – a2)zx = 0, (2.3) BCW': – c2x2 + (b2 – c2)zx + a2yz = 0, (2.4) CAW': b2zx – a2y2 + (c2 – a2)xy = 0, (2.5) ABW': – b2z2 + c2xy + (a2 – b2)yz = 0. (2.6) 3. The Second Brocard triangle PQR The point P = CAW^ABW' has co-ordinates P(b2 + c2 – a2, b2, c2) and it is easily checked that P lies on the Brocard circle. The other two vertices of the Second Brocard triangle have co- ordinates Q(a2, c2 + a2 – b2, c2) and R(a2, b2, a2 + b2 – c2). 4. The Fourth Brocard triangle P'Q'R' The point P' = ABW^CAW' has co-ordinates P'(a2, b2 + c2 – a2, b2 + c2 – a2) and it is easily checked that P' lies on the isogonal locus of the Brocard circle.
[Show full text]
Triangle Circle Limits
The Journal of Symbolic Geometry Volume 1 (2006) Triangle Circle Limits Walker Ray Lakeside High School Lake Oswego, OR [email protected] Abstract: We examine the limit behavior of various triangle circles, called “limit circles”, as the vertices of the triangle coalesce. Introduction To aid my research I used mathematical software programs, most importantly Geometry Expressions and Maple. The use of technology allowed me to complete this research topic successfully. I was able to visualize the problems I was trying to solve and eliminate a great deal of error from my work by utilizing these programs. This allowed me to use creativity in studying my topic and quickly find solutions to the problems I faced. Limit Circles A “limit circle” is an extension of the concept of single dimension interpolation where two or more points have coalesced. The limit circumcircle is a simple, non- trivial case of multidimensional interpolation through points that have coalesced. We will see that the limit circumcircle has some unexpected properties, especially in terms of its relation to the other triangle circles. Limit Circles 25 Limit Circumradius – Two Coalescing Points: C b A a D c B Figure 1. Triangle ABC with circumcircle We start by examining the limit of the circumradius of a triangle as two of the vertices coalesce. To define the radius of the limit circle, we examine the behavior of the circle as vertices B and C approach one another. Using the Law of Sines, we know that the circumradius R of triangle ABC with a sides a, b, and c opposite the angles A, B, and C is R = .
[Show full text]
Lester Circles
International Journal of Computer Discovered Mathematics (IJCDM) ISSN 2367-7775 c IJCDM March 2016, Volume 1, No.1, pp.15-25. Received 15 October 2015. Published on-line 15 December 2015 web: http://www.journal-1.eu/ c The Author(s) This article is published with open access1. Computer Discovered Mathematics: Lester Circles Sava Grozdeva and Deko Dekovb2 a VUZF University of Finance, Business and Entrepreneurship, Gusla Street 1, 1618 Soﬁa, Bulgaria e-mail: [email protected] bZahari Knjazheski 81, 6000 Stara Zagora, Bulgaria e-mail: [email protected] web: http://www.ddekov.eu/ Abstract. A circle is a Lester circle, if it contains at least four remarkable points of the triangle. By using the computer program “Discoverer”, we investigate Lester circles. Keywords. Lester circle, remarkable point, triangle geometry, computer- discovered mathematics, Euclidean geometry, “Discoverer”. Mathematics Subject Classiﬁcation (2010). 51-04, 68T01, 68T99. 1. Introduction From the June Lester web site [8]: “There is always a circle through three given points, as long as they are not on a line. Circles through four given points - those are exceptional. Especially when the four points are well-known special points of a triangle.” June Lester has investigated the circle on which the Circumcenter, Nine-Point Center, and the Outer and Inner Fermat points lie. Now this circle is known as the Lester circle. See [9], [11, Lester Circle], [12, Lester’s theorem]. In 2002 Bernard Gibert [2] discovered that the point Alrescha, that is, point X(1117) in the Kimberling’s ETC [7], also lies on the Lester circle. In 2014 Grozdev and Dekov discovered, with the help of the computer program “Discoverer”, three new remarkable points which lie on the Lester circle.
[Show full text]
On the Invariance of Brocard Angles in the Interior and Exterior Pappus Triangles of Any Given Triangle
Journal for Geometry and Graphics Volume 21 (2017), No. 1, 61{70. On the Invariance of Brocard Angles in the Interior and Exterior Pappus Triangles of any Given Triangle Victor Oxman 1;2, Avi Sigler 1;3, Moshe Stupel 1;4 1Shaanan College, P.O.B. 906, Haifa, Israel email: [email protected] 2Western Galilee College, Acre, Israel 3Technical College for Aviation Professions, Haifa, Israel 4Gordon College, Haifa, Israel Abstract. The paper presents groups of triangles inscribed in a given triangle ABC that have the same Brocard angle as 4ABC . Some of them are similar to 4ABC and some are not. The central group consists of the exterior and interior Pappus triangles of 4ABC . We prove that the Miquel points of Pappus triangles produce pedal triangles that have the same Brocard angle as 4ABC . Furthermore, we prove that the locus of all points, that produce pedal triangles inside 4ABC with the same Brocard angle, is a circle. For Miquel points of exterior Pappus triangles, we prove that all these points are located on the Brocard circle. Key Words: Brocard angles, Brocard points, Brocard circle, Pappus triangles, Miquel points, symmedian point MSC 2010: 51M04, 51M25 1. Brocard points and Brocard angle It is known that every triangle ABC has two Brocard points M1 and M2 (Henry Brocard, French mathematician, 1845{1922). These points are isogonal conjugates (Figure 1), i.e., the lines AM2, BM2 and CM2 are obtained by reﬂecting the lines AM1, BM1 and CM1 in the bisectors of \A, \B, and \C, respectively. The Brocard points satisfy \M1BA = \M1AC = \M1CB = ! and \M2AB = \M2BC = \M2CA = !: The points M1 and M2 are also called the ﬁrst Brocard point and the second Brocard point, respectively.
[Show full text]
The Modern Geometry of the Triangle
H^^fe',- -"*' V^ A W Af* 5Mi (Qantell luiuersity Htbrary Stljaca, New fork BOUGHT WITH THE INCOME OF THE SAGE ENDOWMENT FUND THE GIFT OF HENRY W. SAGE 1891 Mathematics* Cornell University Library QA 482.G16 1910 The modern geometry of the triangle. 3 1924 001 522 782 The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924001522782 THE MODERN GEOMETRY OF THE TRIANGLE. BY WILLIAM GALLATLY, MA. SECOND EDITION. /<?ro London : FRANCIS HODGSON, 89 Faebingdon Street, E.G. •S a. Ms^-rm — PREFACE. In this little treatise on the Geometry of the Triangle are presented some of the more important researches on the subject which have been undertaken during the last thirty years. The author ventures to express not merely his hope, but his con- fident expectation, that these novel and interesting theorems some British, but the greater part derived from French and German sources—will widen the outlook of our mathematical instructors and lend new vigour to their teaching. The book includes some articles contributed by the present writer to the Educational Times Reprint, to whose editor he would offer his sincere thanks for the great encouragement which he has derived from such recognition. He is also most grateful to Sir George Greenhill, Prof. A. C. Dixon, Mr. V. R. Aiyar, Mr. W. F. Beard, Mr. R. F. Davis, and Mr. E. P. Rouse for permission to use the theorems due to them. W. G.
[Show full text]
Triangle and Its Circles from the Standpol T of Modern
View metadata, citationbrought and to similar you by papersCORE at core.ac.uk provided by K-State... A TRIANGLE A*D ITS CIRCLES nCLU MAT COOK Hlrtn ., College, 1913» B.8., Phillips Uhivoraity, 1925 i wmaa submitted In partial fuirillwmt ©f the requlreBenta for the of MatKB OF SCIXSCB XAR8A8 STAK AORICaLTORAL COLLEGE 1920 —— — ——- DoccP rncn+ C65 ttMM OF UWIUM 3 I fetation 4 ELK* BR1ST SBVZSW 0FCIRCLB8 WOM 20 TO A TMAWLB in mrntscr asoBuwr ——— 6 THK TrUASBLB AHD ITS CIRCLES FROM THK STAKDPOT?IT OF QDKM 0K1KETHY 7 Properties of • Triangle used In Connection with Circles Pedsl or Siraeon Line e n s — — 8 8 i —— — . ntlperallela — SjRaaedlon Line end Synwedien Point —— 10 Isogoml Conjugates ~ 12 Appolonius' Theoren lb Circles 18 Pe<:el Circle » — —— lusscribcd and Inscribed -—————-- 21 Zaerlbeu sad Inscribed — — — 28 Hine-polnt Circle » »— 30 "~ 34 P Circle — Feuerbach's Theorem » ' -~ 37 T-emoine's First Circle or Triplicate Satlo 40 Leaoine's Leeond Circle or Cosine Circle ~ Broeard Circle i Circle cr Circle - ?olar Circle «» •s Circle Other Properties of the irlangle lev Um — 71 telner's Point — 74 Terry's Point — it Inversion of a Triangle (a) With Bespeet to Inscribed Circle — 76 ith nespeot to Clrcwsseribod Clrele — 7« (e) If eapeot to Kserlbed Circle -—»-— 77 ACKWOWLKDOFEHT 81 MM Mi 82 ItTRODUCTIOl This dissertation Is an attempt to present a consistent and systematic account of son* of the various theories of the triangle and Its circles from the standpol t of modern geometry end to solve a number of original exercises in connection with these theorem*.
[Show full text]
Barycentric Coordinates Or Barycentrics
Barycentric coordinates or Barycentrics A file of the Geometrikon gallery by Paris Pamfilos The artist finds a greater pleasure in painting than in having completed the picture. Seneca, Letter to Lucilius Contents (Last update: 09‑06‑2021) 1 Preliminaries and definition 2 2 Traces, ratios, harmonic conjugation 3 3 Lines in barycentric coordinates 4 4 Ceva’s and Menelaus’ theorems in barycentrics 5 5 Trilinear polar in barycentrics 6 6 Relation to cartesian coordinates, inner product 8 7 The circumcircle of ABC in barycentrics 9 8 Displacement vectors, inner product, distance 9 9 Orthogonality of lines, orthocentroidal circle 11 10 Distance of a point from a line, distance of two parallels 12 11 Meaning of line coefficients 15 12 Power of a point, general circle, Euler circle 16 13 Centroid, Incenter, Circumcenter, Symmedian point 17 14 Euler line, Orthocenter, center of Euler’s circle 18 15 Triangle Area in Barycentrics 19 16 Circumcevian triangle of a point 20 17 Circle through three points, Brocard circle 21 18 The associated affine transformation 22 19 Relations between barycentrics and cartesian coordinates 25 20 Affine transformations represented in barycentrics 26 21 Remarks on working with barycentrics 28 1 Preliminaries and definition 2 1 Preliminaries and definition “Barycentric coordinates” or “barycentrics” do not use distances of points, but only ratios of lengths and areas. Thus, they belong to the geometry of the “affine plane” ([2, p.191]), which deals with parallels and ratios of lengths of collinear segments without to use a notion of distance between two points, as it does euclidean geometry. In the subsequent discussion points {퐴, 퐵, …} of the plane are identified with two di‑ mensional vectors {(푎1, 푎2), (푏1, 푏2) …}.
[Show full text]