The Symmedian Point
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Orthocorrespondence and Orthopivotal Cubics
Forum Geometricorum b Volume 3 (2003) 1–27. bbb FORUM GEOM ISSN 1534-1178 Orthocorrespondence and Orthopivotal Cubics Bernard Gibert Abstract. We define and study a transformation in the triangle plane called the orthocorrespondence. This transformation leads to the consideration of a fam- ily of circular circumcubics containing the Neuberg cubic and several hitherto unknown ones. 1. The orthocorrespondence Let P be a point in the plane of triangle ABC with barycentric coordinates (u : v : w). The perpendicular lines at P to AP , BP, CP intersect BC, CA, AB respectively at Pa, Pb, Pc, which we call the orthotraces of P . These orthotraces 1 lie on a line LP , which we call the orthotransversal of P . We denote the trilinear ⊥ pole of LP by P , and call it the orthocorrespondent of P . A P P ∗ P ⊥ B C Pa Pc LP H/P Pb Figure 1. The orthotransversal and orthocorrespondent In barycentric coordinates, 2 ⊥ 2 P =(u(−uSA + vSB + wSC )+a vw : ··· : ···), (1) Publication Date: January 21, 2003. Communicating Editor: Paul Yiu. We sincerely thank Edward Brisse, Jean-Pierre Ehrmann, and Paul Yiu for their friendly and valuable helps. 1The homography on the pencil of lines through P which swaps a line and its perpendicular at P is an involution. According to a Desargues theorem, the points are collinear. 2All coordinates in this paper are homogeneous barycentric coordinates. Often for triangle cen- ters, we list only the first coordinate. The remaining two can be easily obtained by cyclically permut- ing a, b, c, and corresponding quantities. Thus, for example, in (1), the second and third coordinates 2 2 are v(−vSB + wSC + uSA)+b wu and w(−wSC + uSA + vSB )+c uv respectively. -
Cevians, Symmedians, and Excircles Cevian Cevian Triangle & Circle
10/5/2011 Cevians, Symmedians, and Excircles MA 341 – Topics in Geometry Lecture 16 Cevian A cevian is a line segment which joins a vertex of a triangle with a point on the opposite side (or its extension). B cevian C A D 05-Oct-2011 MA 341 001 2 Cevian Triangle & Circle • Pick P in the interior of ∆ABC • Draw cevians from each vertex through P to the opposite side • Gives set of three intersecting cevians AA’, BB’, and CC’ with respect to that point. • The triangle ∆A’B’C’ is known as the cevian triangle of ∆ABC with respect to P • Circumcircle of ∆A’B’C’ is known as the evian circle with respect to P. 05-Oct-2011 MA 341 001 3 1 10/5/2011 Cevian circle Cevian triangle 05-Oct-2011 MA 341 001 4 Cevians In ∆ABC examples of cevians are: medians – cevian point = G perpendicular bisectors – cevian point = O angle bisectors – cevian point = I (incenter) altitudes – cevian point = H Ceva’s Theorem deals with concurrence of any set of cevians. 05-Oct-2011 MA 341 001 5 Gergonne Point In ∆ABC find the incircle and points of tangency of incircle with sides of ∆ABC. Known as contact triangle 05-Oct-2011 MA 341 001 6 2 10/5/2011 Gergonne Point These cevians are concurrent! Why? Recall that AE=AF, BD=BF, and CD=CE Ge 05-Oct-2011 MA 341 001 7 Gergonne Point The point is called the Gergonne point, Ge. Ge 05-Oct-2011 MA 341 001 8 Gergonne Point Draw lines parallel to sides of contact triangle through Ge. -
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 M¨obiusintroduced 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. -
Explaining Some Collinearities Amongst Triangle Centres Christopher Bradley
Article: CJB/2011/138 Explaining some Collinearities amongst Triangle Centres Christopher Bradley Abstract: Collinearities are preserved by an Affine Transformation, so that for every collinearity of triangle centres, there is another when an affine transformation has taken place and vice-versa. We illustrate this by considering what happens to three collinearities in the Euclidean plane when an affine transformation takes the symmedian point into the incentre. C' 9 pt circle C'' A B'' Anticompleme ntary triangle F B' W' W E' L M V O X Z E F' K 4 V' K'' Y G T H N' D B U D'U' C Triplicate ratio Circle A'' A' Fig.1 Three collinearities amongst triangle centres in the Euclidean plane 1 1. When the circumconic, the nine-point conic, the triplicate ratio conic and the 7 pt. conic are all circles The condition for this is that the circumconic is a circle and then the tangents at A, B, C form a triangle A'B'C' consisting of the ex-symmedian points A'(– a2, b2, c2) and similarly for B', C' by appropriate change of sign and when this happen AA', BB', CC' are concurrent at the symmedian point K(a2, b2, c2). There are two very well-known collinearities and one less well-known, and we now describe them. First there is the Euler line containing amongst others the circumcentre O with x-co-ordinate a2(b2 + c2 – a2), the centroid G(1, 1, 1), the orthocentre H with x-co-ordinate 1/(b2 + c2 – a2) and T, the nine-point centre which is the midpoint of OH. -
![Arxiv:2101.02592V1 [Math.HO] 6 Jan 2021 in His Seminal Paper [10]](https://docslib.b-cdn.net/cover/7323/arxiv-2101-02592v1-math-ho-6-jan-2021-in-his-seminal-paper-10-957323.webp)
Arxiv:2101.02592V1 [Math.HO] 6 Jan 2021 in His Seminal Paper [10]
International Journal of Computer Discovered Mathematics (IJCDM) ISSN 2367-7775 ©IJCDM Volume 5, 2020, pp. 13{41 Received 6 August 2020. Published on-line 30 September 2020 web: http://www.journal-1.eu/ ©The Author(s) This article is published with open access1. Arrangement of Central Points on the Faces of a Tetrahedron Stanley Rabinowitz 545 Elm St Unit 1, Milford, New Hampshire 03055, USA e-mail: [email protected] web: http://www.StanleyRabinowitz.com/ Abstract. We systematically investigate properties of various triangle centers (such as orthocenter or incenter) located on the four faces of a tetrahedron. For each of six types of tetrahedra, we examine over 100 centers located on the four faces of the tetrahedron. Using a computer, we determine when any of 16 con- ditions occur (such as the four centers being coplanar). A typical result is: The lines from each vertex of a circumscriptible tetrahedron to the Gergonne points of the opposite face are concurrent. Keywords. triangle centers, tetrahedra, computer-discovered mathematics, Eu- clidean geometry. Mathematics Subject Classification (2020). 51M04, 51-08. 1. Introduction Over the centuries, many notable points have been found that are associated with an arbitrary triangle. Familiar examples include: the centroid, the circumcenter, the incenter, and the orthocenter. Of particular interest are those points that Clark Kimberling classifies as \triangle centers". He notes over 100 such points arXiv:2101.02592v1 [math.HO] 6 Jan 2021 in his seminal paper [10]. Given an arbitrary tetrahedron and a choice of triangle center (for example, the circumcenter), we may locate this triangle center in each face of the tetrahedron. -
The Uses of Homogeneous Barycentric Coordinates in Plane Euclidean Geometry by PAUL
The uses of homogeneous barycentric coordinates in plane euclidean geometry by PAUL YIU Department of Mathematical Sciences, Florida Atlantic University, Boca Raton, FL 33431, USA The notion of homogeneous barycentric coordinates provides a pow- erful tool of analysing problems in plane geometry. In this paper, we explain the advantages over the traditional use of trilinear coordinates, and illustrate its powerfulness in leading to discoveries of new and in- teresting collinearity relations of points associated with a triangle. 1. Introduction In studying geometric properties of the triangle by algebraic methods, it has been customary to make use of trilinear coordinates. See, for examples, [3, 6, 7, 8, 9]. With respect to a ¯xed triangle ABC (of side lengths a, b, c, and opposite angles ®, ¯, °), the trilinear coordinates of a point is a triple of numbers proportional to the signed distances of the point to the sides of the triangle. The late Jesuit mathematician Maurice Wong has given [9] a synthetic construction of the point with trilinear coordinates cot ® : cot ¯ : cot °, and more generally, in [8] points with trilinear coordinates a2nx : b2ny : c2nz from one with trilinear coordinates x : y : z with respect to a triangle with sides a, b, c. On a much grandiose scale, Kimberling [6, 7] has given extensive lists of centres associated with a triangle, in terms of trilinear coordinates, along with some collinearity relations. The present paper advocates the use of homogeneous barycentric coor- dinates instead. The notion of barycentric coordinates goes back to MÄobius. See, for example, [4]. With respect to a ¯xed triangle ABC, we write, for every point P on the plane containing the triangle, 1 P = (( P BC)A + ( P CA)B + ( P AB)C); ABC 4 4 4 4 and refer to this as the barycentric coordinate of P . -
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. -
Lemmas in Euclidean Geometry1 Yufei Zhao [email protected]
IMO Training 2007 Lemmas in Euclidean Geometry Yufei Zhao Lemmas in Euclidean Geometry1 Yufei Zhao [email protected] 1. Construction of the symmedian. Let ABC be a triangle and Γ its circumcircle. Let the tangent to Γ at B and C meet at D. Then AD coincides with a symmedian of △ABC. (The symmedian is the reflection of the median across the angle bisector, all through the same vertex.) A A A O B C M B B F C E M' C P D D Q D We give three proofs. The first proof is a straightforward computation using Sine Law. The second proof uses similar triangles. The third proof uses projective geometry. First proof. Let the reflection of AD across the angle bisector of ∠BAC meet BC at M ′. Then ′ ′ sin ∠BAM ′ ′ BM AM sin ∠ABC sin ∠BAM sin ∠ABD sin ∠CAD sin ∠ABD CD AD ′ = ′ sin ∠CAM ′ = ∠ ∠ ′ = ∠ ∠ = = 1 M C AM sin ∠ACB sin ACD sin CAM sin ACD sin BAD AD BD Therefore, AM ′ is the median, and thus AD is the symmedian. Second proof. Let O be the circumcenter of ABC and let ω be the circle centered at D with radius DB. Let lines AB and AC meet ω at P and Q, respectively. Since ∠PBQ = ∠BQC + ∠BAC = 1 ∠ ∠ ◦ 2 ( BDC + DOC) = 90 , we see that PQ is a diameter of ω and hence passes through D. Since ∠ABC = ∠AQP and ∠ACB = ∠AP Q, we see that triangles ABC and AQP are similar. If M is the midpoint of BC, noting that D is the midpoint of QP , the similarity implies that ∠BAM = ∠QAD, from which the result follows. -
The Ballet of Triangle Centers on the Elliptic Billiard
Journal for Geometry and Graphics Volume 24 (2020), No. 1, 79–101. The Ballet of Triangle Centers on the Elliptic Billiard Dan S. Reznik1, Ronaldo Garcia2, Jair Koiller3 1Data Science Consulting Rio de Janeiro RJ 22210-080, Brazil email: [email protected] 2Instituto de Matemática e Estatística, Universidade Federal de Goiás, Goiânia GO, Brazil email: [email protected] 3Universidade Federal de Juiz de Fora, Juiz de Fora MG, Brazil email: [email protected] Abstract. We explore a bevy of new phenomena displayed by 3-periodics in the Elliptic Billiard, including (i) their dynamic geometry and (ii) the non-monotonic motion of certain Triangle Centers constrained to the Billiard boundary. Fasci- nating is the joint motion of certain pairs of centers, whose many stops-and-gos are akin to a Ballet. Key Words: elliptic billiard, periodic trajectories, triangle center, loci, dynamic geometry. MSC 2010: 51N20, 51M04, 51-04, 37-04 1. Introduction We investigate properties of the 1d family of 3-periodics in the Elliptic Billiard (EB). Being uniquely integrable [13], this object is the avis rara of planar Billiards. As a special case of Poncelet’s Porism [4], it is associated with a 1d family of N-periodics tangent to a con- focal Caustic and of constant perimeter [27]. Its plethora of mystifying properties has been extensively studied, see [26, 10] for recent treatments. Initially we explored the loci of Triangle Centers (TCs): e.g., the Incenter X1, Barycenter X2, Circumcenter X3, etc., see summary below. The Xi notation is after Kimberling’s Encyclopedia [14], where thousands of TCs are catalogued. -
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. -
Triangle Centers Defined by Quadratic Polynomials
Math. J. Okayama Univ. 53 (2011), 185–216 TRIANGLE CENTERS DEFINED BY QUADRATIC POLYNOMIALS Yoshio Agaoka Abstract. We consider a family of triangle centers whose barycentric coordinates are given by quadratic polynomials, and determine the lines that contain an infinite number of such triangle centers. We show that for a given quadratic triangle center, there exist in general four principal lines through this center. These four principal lines possess an intimate connection with the Nagel line. Introduction In our previous paper [1] we introduced a family of triangle centers PC based on two conjugates: the Ceva conjugate and the isotomic conjugate. PC is a natural family of triangle centers, whose barycentric coordinates are given by polynomials of edge lengths a, b, c. Surprisingly, in spite of its simple construction, PC contains many famous centers such as the centroid, the incenter, the circumcenter, the orthocenter, the nine-point center, the Lemoine point, the Gergonne point, the Nagel point, etc. By plotting the points of PC, we know that they are not independently distributed in R2, or rather, they possess a strong mutual relationship such as 2 : 1 point configuration on the Euler line (see Figures 2, 4). In [1] we generalized this 2 : 1 configuration to the form “generalized Euler lines”, which contain an infinite number of centers in PC . Furthermore, the centers in PC seem to have some additional structures, and there frequently appear special lines that contain an infinite number of centers in PC. In this paper we call such lines “principal lines” of PC. The most famous principal lines are the Euler line mentioned above, and the Nagel line, which passes through the centroid, the incenter, the Nagel point, the Spieker center, etc. -
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.