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The Modern Geometry of the Triangle

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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. — — :: SYNOPSIS OF CHAPTERS. [The numbers refer to Sections.] CHAPTEB I. —DlBECTION ANGLES. (1-13.) Relations between 6 X 63 93 and pqr : lines at rt. angles, condition: re- lation between pqr : distance between (0^7) and (a'ff'y') : perpr. on line : Properties of Quad. Centre Circle. Chapteb II.—Medial and Tbipolab Coobdinates. (14-21.) Medial formulas : Peuerbach Point : mid-point line of quad. : N.P. Circle : Tripolar eqn. to circumdiameter : straight line (general) circle : to find points with given trip. c. : trip. c. of Limiting Points. Chapter III. Pobistic Triangles. (22-34.) Poristic condition : R.A. of 0(R) and I(r) : points S) S^trHiGi poristically fixed : circular loci of FGHOTNM : properties of Nagel Point N Gergonne Point M : poristic formulas : locus of M found. Chapter IV. Simson Lines. (35-55.) Draw S.L. in given direction : S.L. bisects TH: OAT = a x : perps. from A, B, C on S.L. : eqn. to S.L. : perpr. TV, length and direction : the Niov quad. : pairs of S.L. the point : circle (kkf) : to point a eqn. TOT' ; ABC and A'B'C n.c. of w : the centre Oi of (kh') : tsicusp Hypocycloid : Cubic in sin</> or cos* : Greenhill and Dixon's Theorem. Chapteb V.—Pedal Triangles. (56-69.) Pedal triangle def similar to LMN : B8C = A + k : sin A a art : Limiting Points : n in terms of xyz : Radical Axes (group) : Feuerbach's Theorem : 2 : Artzt's : V, n, OS , and (a&y) in terms of A^c Parabola So for S lt inverse of : of T, T' are axes of similitude for e . S S.L. def, d1 1 fi — ——— : VI MODERN GEOMETRY. Chapter VI. The Obthopole. (70-80.) geo- Concurrence of pp', qq', rr' : S = 2B cos 9, cos »2 cos e3 ; S found metrically : general n.c. of S : point <r lies on N.P. circle : ABC and A'B'C n.c. of a : a coincides with a : three S.L. through a point, reciprocal relations : Lemoyne's Theorem : pedal circles through a (as) : constant b.c. of S for def: Harmonic system of lines, a common to the 4 circles. (See Appendix I and II.) Chapter VII. Antipedal Triangles. (81-87.) Angular c. : Orthologic and Antipedal triangles : no. of S' : V = \M : n.c. of centre of similitude of U and V : similar properties of Si and S\ : 8' Vi— V\ = 4a ; and S\ are called Twin Points. Chapter VIII. Orthogonal Projection of a Triangle. 88-100.) Shape and size of projection of ABC on plane passipg through fixed axis in plane of ABC : ABC projected into triangle with given angles : projec- 2 2 tion of ABC on planes at constant inclination : 2o' cot A = 2 A (1 + cos «)...: equilat. triangle and Brocard Angle : Antipedal triangles and projection : calculation of a', b', c' : Pedal triangles and projection : general theory for any triangle XYZ : Sehoute Circles. Chapter IX. Counter Points. (101-116.) 2 2 a', II', D', M', q' in terms of \/j.v : nn' = 4E g : equation to minor axis: SA. S'A = AB.Ac : A'B'C b.c. of cr : p- = Ja/JV : Ratio Bl : CI pedal circle SS' cuts N.P. circle at a,u : Aiyar's Theorem, OS.OS' = 2iJ.OV : M'Cay's Cubic : Counterpoint conies, direction of asymptotes and axes : 2 For TOT, conic is R.H. ; centre, asymptotes, (semi-axis) : similarly for S' and S'i : join of Twin Points S] S\ bisected at a ; circle centre S radius SS\ passes through I'm'n'. Chapter X.—Lemoine Geometry. (117-129.) K found : n.c. and b.c. of of K: E centroid def: equation to OK: n : 2 2 2 2 2 2 a + j8 + 7 and m + + «> each a min. at if: Artzt's Parabola: triangle " " TiTs T3 : Lemoine Point of IiI2 I3 : list of {s-a) points : AK bisects chords parallel : to T2 T3 Harmonic quad. A'l passes through K : locus 2 c'2 of centres of rect. inscribed in ABC : cos0! a a(6 — ) : tripolar equation to OK ; Apollonian Circles : Lemoine Axis : Harmonic quad, inverts into square. —— : SYNOPSIS OF CHAPTERS. Vll Chapter XI. Lemoine-Brocard Geometry. (130-152.) Forms for cot a : n.o. and b.c. of n and n' : equation to an' : a not greater than 30° : useful formulse : Neuberg Circles : Steiner Angles Pedal triangle of a : Triangles XBC, YCA, ZAB (see also Appendix III) : First Brooard Triangle PQB : centre of perspective D : eqn. to axis of perspective : G double point of ABC, PQB : OK bisects nn' at right angles : On = eB, OK = eB sec a> : Steiner and Tarry Points : figure KPBOQ similar to 2ACTB: O mean centre of points yzx, nxy, xyz: apply to PQB ; G oentroid of Dan' : D lies on 2 OT : OD = e-B : 3' Isodynamic Points S and 5, : Isogonic points and S,' : Circum-Ellipse and Steiner Ellipse. Chapter XII. Pivot Points. Tucker Circles. (153-166.) Pairs of homothetic triangles, inscribed and circumscribed to ABC : family of circles touch conic : Tucker Circles : list of formulse : Radical Axis : First Lemoine Circle : Pedal circle of aa' : Second Lemoine Circle : Taylor Circle : trip. c. of Limiting Points for Taylor Circle are as cot A, cotS, cot C. CHAPTER I. INTRODUCTION : DIRECTION" ANGLES. 1. In this work the following conventions are observed:—the Circumoentre of ABG, the triangle of reference, will be denoted by ; the Orthocentre by M; the feet of the perpendiculars from A, B, G on BG, CA, AB respectively by H H , H (the it 2 s triangle ffjffjfl, being called the Orthocentric Triangle) ; the h or lengths of AMV BH„, CHS by lt h% , h3 ; the Centroid Centre I, J,, of Gravity by G ; the in- and ex-centres by J2 , Js ; the points of contact of the circle I with the sides of ABG by X, Y, Z ; the corresponding points for the circle I, being X„ x„ Z\. The Circumcentre of I-J^ is «7j 'which lies on 01; also OJ = 01, and the Circumradius of J^Jg = 2B. B . 2 MODERN GEOMETRY. The mid-points of BO, GA, AB are A', B', C ; the triangle A'B'C being called the Medial Triangle. Its Circumcircle is 0' the Nine-Point Circle, with Centre and radius = \R ; the Orthocentre of A'B'C is 0, the Centroid is G, while the in-centre is denoted by 1'. The lines drawn through A, B, parallel to BO, OA, AB form the Anti-Medial Triangle A-fi-fix. Its Circumcentre is H, its Centroid is G, its Medial or Nine- Point Circle is ABO ; and its Nine-Point Centre is 0. The letters " n.c." stand for normal or trilinear coordinates denoted by a/3y. The letters " b.c." stand for barycentric or areal or triangular coordinates, denoted by xyz ; also x = aa, &c, so that x + y+z = 2X 2. Let p, q, r be the lengths of the perpendiculars from A,B, on a straight line 2'2" Let $x, #2 , a be the Direction Angles of TT' ; that is, the angles which the sides of ABO make with TT', these angles being measured from TT' as axis, and in the same sense. By projecting the sides of ABO along and perpendicular to TT', we obtain 2 . a cos 0, = 2.asin(9 0, 1 =0. The diagram shows that a sin 6 = q— &c, so that X r, 2 ap sin &x= %.p (q—r) = 0. Two sets of direction angles should be particularly noted. For 01, cos Ox = A'X/OI = \ (b-c)/0I oc (6-c). For OGH, cos 6x = A'HxfOH = li ain(B-C)/0H cc (tf-c^/a. DIRECTION ANGLES. 3 To express cos 6 in terms r. X of p, q, Since Off, : H B = b cos . c cos B, X .. q.b cos G+r.c cos B = (& cos 0+c cos P)!?^ = a(p — H d') = ap— a.AH cos6 1 l l ; .'. 2A.cos 6, = ap— 6g cos — or cos B; When ff, falls outside dd', the right-hand signs are changed.
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  • MYSTERIES of the EQUILATERAL TRIANGLE, First Published 2010

    MYSTERIES of the EQUILATERAL TRIANGLE, First Published 2010

    MYSTERIES OF THE EQUILATERAL TRIANGLE Brian J. McCartin Applied Mathematics Kettering University HIKARI LT D HIKARI LTD Hikari Ltd is a publisher of international scientific journals and books. www.m-hikari.com Brian J. McCartin, MYSTERIES OF THE EQUILATERAL TRIANGLE, First published 2010. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission of the publisher Hikari Ltd. ISBN 978-954-91999-5-6 Copyright c 2010 by Brian J. McCartin Typeset using LATEX. Mathematics Subject Classification: 00A08, 00A09, 00A69, 01A05, 01A70, 51M04, 97U40 Keywords: equilateral triangle, history of mathematics, mathematical bi- ography, recreational mathematics, mathematics competitions, applied math- ematics Published by Hikari Ltd Dedicated to our beloved Beta Katzenteufel for completing our equilateral triangle. Euclid and the Equilateral Triangle (Elements: Book I, Proposition 1) Preface v PREFACE Welcome to Mysteries of the Equilateral Triangle (MOTET), my collection of equilateral triangular arcana. While at first sight this might seem an id- iosyncratic choice of subject matter for such a detailed and elaborate study, a moment’s reflection reveals the worthiness of its selection. Human beings, “being as they be”, tend to take for granted some of their greatest discoveries (witness the wheel, fire, language, music,...). In Mathe- matics, the once flourishing topic of Triangle Geometry has turned fallow and fallen out of vogue (although Phil Davis offers us hope that it may be resusci- tated by The Computer [70]). A regrettable casualty of this general decline in prominence has been the Equilateral Triangle. Yet, the facts remain that Mathematics resides at the very core of human civilization, Geometry lies at the structural heart of Mathematics and the Equilateral Triangle provides one of the marble pillars of Geometry.