Introduction to the Geometry of the Triangle
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Investigating Centers of Triangles: the Fermat Point
Investigating Centers of Triangles: The Fermat Point A thesis submitted to the Miami University Honors Program in partial fulfillment of the requirements for University Honors with Distinction by Katherine Elizabeth Strauss May 2011 Oxford, Ohio ABSTRACT INVESTIGATING CENTERS OF TRIANGLES: THE FERMAT POINT By Katherine Elizabeth Strauss Somewhere along their journey through their math classes, many students develop a fear of mathematics. They begin to view their math courses as the study of tricks and often seemingly unsolvable puzzles. There is a demand for teachers to make mathematics more useful and believable by providing their students with problems applicable to life outside of the classroom with the intention of building upon the mathematics content taught in the classroom. This paper discusses how to integrate one specific problem, involving the Fermat Point, into a high school geometry curriculum. It also calls educators to integrate interesting and challenging problems into the mathematics classes they teach. In doing so, a teacher may show their students how to apply the mathematics skills taught in the classroom to solve problems that, at first, may not seem directly applicable to mathematics. The purpose of this paper is to inspire other educators to pursue similar problems and investigations in the classroom in order to help students view mathematics through a more useful lens. After a discussion of the Fermat Point, this paper takes the reader on a brief tour of other useful centers of a triangle to provide future researchers and educators a starting point in order to create relevant problems for their students. iii iv Acknowledgements First of all, thank you to my advisor, Dr. -
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. -
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. -
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. -
Degree of Triangle Centers and a Generalization of the Euler Line
Beitr¨agezur Algebra und Geometrie Contributions to Algebra and Geometry Volume 51 (2010), No. 1, 63-89. Degree of Triangle Centers and a Generalization of the Euler Line Yoshio Agaoka Department of Mathematics, Graduate School of Science Hiroshima University, Higashi-Hiroshima 739–8521, Japan e-mail: [email protected] Abstract. We introduce a concept “degree of triangle centers”, and give a formula expressing the degree of triangle centers on generalized Euler lines. This generalizes the well known 2 : 1 point configuration on the Euler line. We also introduce a natural family of triangle centers based on the Ceva conjugate and the isotomic conjugate. This family contains many famous triangle centers, and we conjecture that the de- gree of triangle centers in this family always takes the form (−2)k for some k ∈ Z. MSC 2000: 51M05 (primary), 51A20 (secondary) Keywords: triangle center, degree of triangle center, Euler line, Nagel line, Ceva conjugate, isotomic conjugate Introduction In this paper we present a new method to study triangle centers in a systematic way. Concerning triangle centers, there already exist tremendous amount of stud- ies and data, among others Kimberling’s excellent book and homepage [32], [36], and also various related problems from elementary geometry are discussed in the surveys and books [4], [7], [9], [12], [23], [26], [41], [50], [51], [52]. In this paper we introduce a concept “degree of triangle centers”, and by using it, we clarify the mutual relation of centers on generalized Euler lines (Proposition 1, Theorem 2). Here the term “generalized Euler line” means a line connecting the centroid G and a given triangle center P , and on this line an infinite number of centers lie in a fixed order, which are successively constructed from the initial center P 0138-4821/93 $ 2.50 c 2010 Heldermann Verlag 64 Y. -
The Feuerbach Point and the Fuhrmann Triangle
Forum Geometricorum Volume 16 (2016) 299–311. FORUM GEOM ISSN 1534-1178 The Feuerbach Point and the Fuhrmann Triangle Nguyen Thanh Dung Abstract. We establish a few results on circles through the Feuerbach point of a triangle, and their relations to the Fuhrmann triangle. The Fuhrmann triangle is perspective with the circumcevian triangle of the incenter. We prove that the perspectrix is the tangent to the nine-point circle at the Feuerbach point. 1. Feuerbach point and nine-point circles Given a triangle ABC, we consider its intouch triangle X0Y0Z0, medial trian- gle X1Y1Z1, and orthic triangle X2Y2Z2. The famous Feuerbach theorem states that the incircle (X0Y0Z0) and the nine-point circle (N), which is the common cir- cumcircle of X1Y1Z1 and X2Y2Z2, are tangent internally. The point of tangency is the Feuerbach point Fe. In this paper we adopt the following standard notation for triangle centers: G the centroid, O the circumcenter, H the orthocenter, I the incenter, Na the Nagel point. The nine-point center N is the midpoint of OH. A Y2 Fe Y0 Z1 Y1 Z0 H O Z2 I T Oa B X0 U X2 X1 C Ja Figure 1 Proposition 1. Let ABC be a non-isosceles triangle. (a) The triangles FeX0X1, FeY0Y1, FeZ0Z1 are directly similar to triangles AIO, BIO, CIO respectively. (b) Let Oa, Ob, Oc be the reflections of O in IA, IB, IC respectively. The lines IOa, IOb, IOc are perpendicular to FeX1, FeY1, FeZ1 respectively. Publication Date: July 25, 2016. Communicating Editor: Paul Yiu. 300 T. D. Nguyen Proof. -
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. -
From Euler to Ffiam: Discovery and Dissection of a Geometric Gem
From Euler to ffiam: Discovery and Dissection of a Geometric Gem Douglas R. Hofstadter Center for Research on Concepts & Cognition Indiana University • 510 North Fess Street Bloomington, Indiana 47408 December, 1992 ChapterO Bewitched ... by Circles, Triangles, and a Most Obscure Analogy Although many, perhaps even most, mathematics students and other lovers of mathematics sooner or later come across the famous Euler line, somehow I never did do so during my student days, either as a math major at Stanford in the early sixties or as a math graduate student at Berkeley for two years in the late sixties (after which I dropped out and switched to physics). Geometry was not by any means a high priority in the math curriculum at either institution. It passed me by almost entirely. Many, many years later, however, and quite on my own, I finally did become infatuated - nay, bewitched- by geometry. Just plane old Euclidean geometry, I mean. It all came from an attempt to prove a simple fact about circles that I vaguely remembered from a course on complex variables that I took some 30 years ago. From there on, the fascination just snowballed. I was caught completely off guard. Never would I have predicted that Doug Hofstadter, lover of number theory and logic, would one day go off on a wild Euclidean-geometry jag! But things are unpredictable, and that's what makes life interesting. I especially came to love triangles, circles, and their unexpectedly profound interrelations. I had never appreciated how intimately connected these two concepts are. Associated with any triangle are a plentitude of natural circles, and conversely, so many beautiful properties of circles cannot be defined except in terms of triangles. -
The New Proof of Euler's Inequality Using Spieker Center
of Math al em rn a u ti o c International Journal of Mathematics And its Applications J s l A a n n d o i i Volume 3, Issue 4{E (2015), 67{73. t t a s n A r e p t p ISSN: 2347-1557 n l I i c • a t 7 i o 5 n 5 • s Available Online: http://ijmaa.in/ 1 - 7 4 I 3 S 2 S : N International Journal of Mathematics And its Applications The New Proof of Euler's Inequality Using Spieker Center Research Article Dasari Naga Vijay Krishna1∗ 1 Department of Mathematics, Keshava Reddy Educational Instutions, Machiliaptnam, Kurnool, India. Abstract: If R is the Circumradius and r is the Inradius of a non-degenerate triangle then due to EULER we have an inequality referred as \Euler's Inequality"which states that R ≥ 2r, and the equality holds when the triangle is Equilateral. In this article let us prove this famous inequality using the idea of `Spieker Center '. Keywords: Euler's Inequality, Circumcenter, Incenter, Circumradius, Inradius, Cleaver, Spieker Center, Medial Triangle, Stewart's Theorem. c JS Publication. 1. Historical Notes [4] In 1767 Euler analyzed and solved the construction problem of a triangle with given orthocenter, circum center, and in center. The collinearlity of the Centroid with the orthocenter and circum center emerged from this analysis, together with the celebrated formula establishing the distance between the circum center and the in center of the triangle. The distance d between the circum center and in center of a triangle is given by d2 = R(R − 2r) where R, r are the circum radius and in radius, respectively. -
Barycentric Coordinates: Formula Sheet
International Journal of Computer Discovered Mathematics (IJCDM) ISSN 2367-7775 c IJCDM June 2016, Volume 1, No.2, pp.75-82. Received 20 March 2016. Published on-line 20 April 2016 web: http://www.journal-1.eu/ c The Author(s) This article is published with open access1. Barycentric Coordinates: Formula Sheet 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. We present several basic formulas about the barycentric coordinates. The aim of the formula sheet is to serve for references. Keywords. barycentric coordinates, triangle geometry, notable point, Euclidean geometry. Mathematics Subject Classification (2010). 51-04, 68T01, 68T99. 1. Area, Points, Lines We present several basic formulas about the barycentric coordinates. The aim of the formula sheet is to serve for references. We refer the reader to [15],[2],[1],[8],[3],[6],[7],[10],[11],[9],[12]. The labeling of triangle centers follows Kimberling’s ETC [8]. The reader may find definitions in [13],[14], [5, Contents, Definitions]. The reference triangle ABC has vertices A = (1; 0; 0);B(0; 1; 0) and C(0,0,1). The side lengths of 4ABC are denoted by a = BC; b = CA and c = AB. A point is an element of R3, defined up to a proportionality factor, that is, For all k 2 R − f0g : P = (u; v; w) means that P = (u; v; w) = (ku; kv; kw): Given a point P (u; v; w). -
Arxiv:2012.11270V3 [Math.MG] 13 Jul 2021 Altaetr Emnsaetnett Ofclcaustic) Confocal a to Tangent Are Segments Trajectory (All I)Invariants
FAMILY TIES: RELATING PONCELET 3-PERIODICS BY THEIR PROPERTIES RONALDO GARCIA AND DAN REZNIK Abstract. We compare loci types and invariants across Poncelet families interscribed in three distinct concentric Ellipse pairs: (i) ellipse-incircle, (ii) circumcircle-inellipse, and (iii) homothetic. Their metric properties are mostly identical to those of 3 well-studied families: elliptic billiard (confocal pair), Chapple’s poristic triangles, and the Brocard porism. We therefore organized them in three related groups. Keywords invariant, elliptic, billiard, locus. MSC 51M04 and 51N20 and 51N35 and 68T20 1. Introduction We have been studying loci and invariants of Poncelet 3-periodics in the confocal ellipse pair (elliptic billiard). Classic invariants include Joachmisthal’s constant J (all trajectory segments are tangent to a confocal caustic) and perimeter L [26]. A few properties detected experimentally [21] and later proved can be divided into two groups: (i) loci of triangle centers (we use the Xk notation in [17]), and (ii) invariants. In terms of loci, the following results have been proved: (i) the locus of the incenter [9, 23], barycenter [25], circumcenter [7, 9], orthocenter [11] and many others are ellipses; (ii) a special triangle center known as the Mittenpunkt X9 is stationary [24]. For invariants we chiefly have (i) the sum of cosines [1, 2], (ii) the product of outer polygon cosines, and (iii) outer-to-3-periodic area ratio [4]. We continue our inquiry into loci and invariants by now considering 3-periodic families three other non-confocal though concentric ellipse pairs. Referring to Fig- arXiv:2012.11270v3 [math.MG] 13 Jul 2021 ure 1: Family I: outer ellipse and incircle, incenter X1 is stationary. -
Poristic Loci of Triangle Centers
Journal for Geometry and Graphics Volume 15 (2011), No. 1, 45–67. Poristic Loci of Triangle Centers Boris Odehnal Institute of Discrete Mathematics and Geometry, Vienna University of Technology Wiedner Hauptstr. 8-10/104, A 1040 Vienna, Austria email: [email protected] Abstract. The one-parameter family of triangles with common incircle and circumcircle is called a porisitic1 system of triangles. The triangles of a poristic system can be rotated freely about the common incircle. However this motion is not a rigid body motion for the sidelengths of the triangle are changing. Sur- prisingly many triangle centers associated with the triangles of the poristic family trace circles while the triangle traverses the poristic family. Other points move on conic sections, some points trace more complicated curves. We shall describe the orbits of centers and some other points. Thereby we are able to answer open questions and verify some older results. Key Words: Poristic triangles, incircle, excircle, non-rigid body motion, poristic locus, triangle center, circle, conic section MSC 2010: 51M04, 51N35 1. Introduction The family of poristic triangles has marginally attracted geometers interest. There are only a few articles contributing to this particular topic of triangle geometry: [3] is dedicated to perspective poristic triangles, [12] deals with the existence of triangles with prescribed circumcircle, incircle, and an additional element. Some more general appearances of porisms are investigated in [2, 4, 6, 9, 16] and especially [7] provides an overview on Poncelet’s theorem which is the projective version and thus more general notion of porism. Nevertheless there are some results on poristic loci, i.e., the traces of triangle centers and other points related to the triangle while the triangle is traversing the poristic family.