DOCSLIB.ORG

Nagel Point of the Triangle by Paris Pamfilos

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Nagel Point of the Triangle by Paris Pamfilos Nagel point of the triangle by Paris Pamfilos The bottom line for mathematicians is that the architecture has to be right. In all the mathematics that I did, the essential point was to find the right architecture. It’s like building a bridge. Once the main lines of the structure are right, then the details miraculously fit. The problem is the overall design. C.L. Dodgson, College Math. J. 25(1994) Contents 1 Nagel point of the triangle1 2 Barycentric coordinates of the Nagel point2 3 The Nagel line of the triangle2 4 Alternative construction of the Nagel point3 5 Other Nagel-like points3 6 Connection with the de Longchamps point4 1 Nagel point of the triangle This is defined as the intersection point Na of the lines joining the vertices fA; B;Cg with the contact points fA00; B00;C00g of the opposite sides with the corresponding “excircles” of the triangle t = ABC: That this point exists can be easily proved by applying “Ceva’s theorem” (see file Ceva’s theorem) A N a B A'' C C' B' O A A00B s−c Figure 1: Nagel’s point characteristic ratios A00C = − s−b 2 Barycentric coordinates of the Nagel point 2 Theorem 1. The lines fAA00; BB00;CC00g intersect at a point. This follows by measuring the ratio and applying Ceva’s theorem A00B BB0 s − c A00B B00C C00 A ¹s − cº¹s − aº¹s − bº = − = − ) · · = − = −1; A00C CC0 s − b A00C B00 A C00B ¹s − bº¹s − cº¹s − aº where fa = jBCj; b = jCAj; c = jABjg the side-lengths and s = ¹a + b + c)/2 the half-perimeter of the triangle, s = jAB0j = jAC0j (See Figure 1). 2 Barycentric coordinates of the Nagel point For the calculation of the barycentrics (see file Barycentric coordinates) of Na we use the known signed ratio A00B s − c 1 1 r = = − ) A00 = ¹B − rCº ) A00 = ¹¹s − bºB + ¹s − cºCº: A00C s − b 1 − r a 00 1 Analogously we obtain B = b ¹¹s − cºC + ¹s − aºAº: The Nagel point is the intersection of 00 00 the lines Na = AA \ BB ; the coefficients of which are expressed by the vector products AA00 : A × A00 = ¹0 : −(s − cº : ¹s − bºº; BB00 : B × B00 = ¹¹s − cº : 0 : −(s − aºº: The barycentrics of Na result by taking again the vector product AA00 × BB00 = ¹s − a : s − b : s − cº: 3 The Nagel line of the triangle This is the line containing the incenter, the centroid and the Nagel point and, as Bottema says, it is “a counterpart of the Euler line” [Bot07, p.83]. The justification for this is given by the following theorem. 0 0 0 0 Theorem 2. The Nagel point Na is the incenter of the “anticomplementary” triangle t = A B C of ABC: The points fI;G; Nag are collinear and jGNa j = 2jIGj: Consider the homothety f with center G and ratio −2: This maps the triangle t onto t0 and the incircle κ of t onto the incircle κ0 of t0: The proof amounts to show 0 0 that the line A1I passes through the vertex A (See Figure 2). Here fI; I ; A2; A1g denote respectively the incenters of t;t0 and the contact points of κ with fBC; DEg, where DE is the parallel to BC tangent to κ: 0 0 The triangle A1I A2 has G for centroid. Hence the line A2 A3 = f ¹I A1º: Because of 00 0 0 0 0 BA2 = CA = s − b; next ratios in the similar triangles fADE; A CB; A D E g are equal: 0 A1E A2B A3E = = 0 : A1D A2C A3D 0 0 Hence A2 A3 passes through A and consequently its homothetic I A1 passes through A: By the similarity of triangles fADE; ABCg line AI 0 passes also through the contact point A00 of the excircle with BC . Thus, the cevian AA00 containing the Nagel point passes through I 0 and analogously the other cevians do the same. Hence I 0 coincides with the Nagel point Na of ABC: 4 Alternative construction of the Nagel point 3 A A C' 4 B' D A1 E I G I'(Na) κ B C A2 A'' κ' E' D' A3 A' Figure 2: The cevians of the Nagel point 4 Alternative construction of the Nagel point The following method ([Ask03, p.13]), rediscovered in [Hoe07]) gives another way to construct the Nagel point using only the incircle and not the excircles. For this draw tangents to the incircle parallel to the sides. Then join the contact points of these parallels with the opposite vertices. The three cevians thus created concure at the Nagel point (See A D E U I V N a B C Figure 3: Alternative definition of the Nagel point Figure 3). The proof follows directly from the similarity of triangles fADE; ABCg alluded to also in the previous section. 5 Other Nagel-like points Below is drawn an extension of the figure1 defining the Nagel point. In this there are seen three additional Nagel-like points fN1; N2; N3g, resulting as intersections of three cevians to the contact points with the “tritangent circles” of the triangle. 1. If point U is the contact of incircle with side AB; then its antipode V is on the line CNa: 2. The extension of CB1 passes through the antipode of A1: 6 Connection with the de Longchamps point 4 B 1 C 1 B'' A C'' N A 3 N 3 2 C' B' N U a I V A' C B C B 2 2 N 1 A 2 A 1 A'' B 3 Figure 4: Other Nagel-like points fN1; N2; N3g Nr-1 is a consequence of the homothety of the incircle to the excircle opposite to C: The homothety has center at C and this implies that the end-points of parallel radii of the two circles are aligned with C: This is the case with fC0;Vg. Nr-2 holds for, essentially, the same reason. This time C is the (anti) homothety center of the two excircles with centers A”, B”, hence again end-points of anti-parallel radii are collinear with C. This is the case with fB1; B3g. Analogous properties, of course, hold also for the other vertices of the triangle and the corresponding excircles and contacts. 6 Connection with the de Longchamps point The “de Longchamps” point of the triangle is the symmetric of the orthocenter w.r. to the circumcenter of the triangle. Next theorem relates it to the Nagel-like points of the triangle. 00 00 00 Theorem 3. The three Nagel-like points fN1; N2; N3g joined to respective excenters fA ; B ;C g define three concurring lines at the “de Longchamps” point De of the triangle. Thus De is the center of perspectivity of the two triangles fABC; N1N2N3g and coin- cides with the (see file De Longchamps point). (See Figure 5). As we did in section2, we compute here also the barycentrics of the contact points from the known ratios, the symbol denoting here the equality of vectors up to a non- 6 Connection with the de Longchamps point 5 B B'' 1 C 1 A D e N C'' 3 C' N 2 B' N I a A' B C B C 2 2 N A 1 2 A A'' 1 Figure 5: de Longchamps point related to fN1; N2; N3g zero scalar factor: A1 A s 1 = r = ) A1 = ¹A − rBº ¹¹c − sºA + sBº; A1B s − c 1 − r A2C s − b 1 = r = ) A2 = ¹A − r Aº ¹sC + ¹b − cºAº; A2 A s 1 − C A1C : A1 × C = (−s : c − s : 0º; A2B : A2 × B = ¹s : 0 : s − bº ) N1 = ¹¹s − bº¹c − sº : s¹s − bº : s¹s − cºº: The collinearity of fA; Na; N1g and the similar to it triples, suggested by figure 5, results from the obviously vanishing determinant 1 0 0 s − a s − b s − c = 0: ¹s − bº¹c − sº s¹s − bº s¹s − cº Similar to the previous arguments lead to the barycentrics A00 = (−a : b : cº; B00 = ¹a : −b : cº; C00 = ¹a : b : −cº: Taking into account that the de Longchamps point has barycentrics 4 2 2 2 2 2 2 De = (−3a + 2a ¹b + c º + ¹b + c º ;::: º 00 and computing the determinant of the barycentrics vectors of the points fDe; N1; A g we find that this vanishes, hence the three points are collinear. This proves that the three lines 00 00 00 fA N1; B N2;C N3g pass through the de Longchamps point De of the triangle ABC: BIBLIOGRAPHY 6 Bibliography [Ask03] E.H. Askwith. A course of pure Geometry. Cambridge University Press, Cam- bridge, 1903. [Bot07] Oene Bottema. Topics in Elementary Geometry. Springer Verlag, Heidelberg, 2007. [Hoe07] L. Hoehn. A New Characterization of the Nagel Point. Missouri J. Math. Sci., 19:45–48, 2007..
Load more

Recommended publications
  • Relationships Between Six Excircles
    Sangaku Journal of Mathematics (SJM) c SJM ISSN 2534-9562 Volume 3 (2019), pp.73-90 Received 20 August 2019. Published on-line 30 September 2019 web: http://www.sangaku-journal.eu/ c The Author(s) This article is published with open access1. Relationships Between Six Excircles Stanley Rabinowitz 545 Elm St Unit 1, Milford, New Hampshire 03055, USA e-mail: [email protected] web: http://www.StanleyRabinowitz.com/ Abstract. If P is a point inside 4ABC, then the cevians through P divide 4ABC into smaller triangles of various sizes. We give theorems about the rela- tionship between the radii of certain excircles of some of these triangles. Keywords. Euclidean geometry, triangle geometry, excircles, exradii, cevians. Mathematics Subject Classification (2010). 51M04. 1. Introduction Let P be any point inside a triangle ABC. The cevians through P divide 4ABC into six smaller triangles. In a previous paper [5], we found relationships between the radii of the circles inscribed in these triangles. For example, if P is at the orthocenter H, as shown in Figure 1, then we found that r1r3r5 = r2r4r6, where the ri are radii of the incircles as shown in the figure. arXiv:1910.00418v1 [math.HO] 28 Sep 2019 Figure 1. r1r3r5 = r2r4r6 In this paper, we will find similar results using excircles instead of incircles. When the cevians through a point P interior to a triangle ABC are drawn, many smaller 1This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
    [Show full text]
  • The Stammler Circles
    Forum Geometricorum b Volume 2 (2002) 151–161. bbb FORUM GEOM ISSN 1534-1178 The Stammler Circles Jean-Pierre Ehrmann and Floor van Lamoen Abstract. We investigate circles intercepting chords of specified lengths on the sidelines of a triangle, a theme initiated by L. Stammler [6, 7]. We generalize his results, and concentrate specifically on the Stammler circles, for which the intercepts have lengths equal to the sidelengths of the given triangle. 1. Introduction Ludwig Stammler [6, 7] has investigated, for a triangle with sidelengths a, b, c, circles that intercept chords of lengths µa, µb, µc (µ>0) on the sidelines BC, CA and AB respectively. He called these circles proportionally cutting circles,1 and proved that their centers lie on the rectangular hyperbola through the circumcenter, the incenter, and the excenters. He also showed that, depending on µ, there are 2, 3 or 4 circles cutting chords of such lengths. B0 B A0 C A C0 Figure 1. The three Stammler circles with the circumtangential triangle As a special case Stammler investigated, for µ =1, the three proportionally cutting circles apart from the circumcircle. We call these the Stammler circles. Stammler proved that the centers of these circles form an equilateral triangle, cir- cumscribed to the circumcircle and homothetic to Morley’s (equilateral) trisector Publication Date: November 22, 2002. Communicating Editor: Bernard Gibert. 1Proportionalschnittkreise in [6]. 152 J.-P. Ehrmann and F. M. van Lamoen triangle. In fact this triangle is tangent to the circumcircle at the vertices of the circumtangential triangle. 2 See Figure 1. In this paper we investigate the circles that cut chords of specified lengths on the sidelines of ABC, and obtain generalizations of results in [6, 7], together with some further results on the Stammler circles.
    [Show full text]
  • Special Isocubics in the Triangle Plane
    Special Isocubics in the Triangle Plane Jean-Pierre Ehrmann and Bernard Gibert June 19, 2015 Special Isocubics in the Triangle Plane This paper is organized into five main parts : a reminder of poles and polars with respect to a cubic. • a study on central, oblique, axial isocubics i.e. invariant under a central, oblique, • axial (orthogonal) symmetry followed by a generalization with harmonic homolo- gies. a study on circular isocubics i.e. cubics passing through the circular points at • infinity. a study on equilateral isocubics i.e. cubics denoted 60 with three real distinct • K asymptotes making 60◦ angles with one another. a study on conico-pivotal isocubics i.e. such that the line through two isoconjugate • points envelopes a conic. A number of practical constructions is provided and many examples of “unusual” cubics appear. Most of these cubics (and many other) can be seen on the web-site : http://bernard.gibert.pagesperso-orange.fr where they are detailed and referenced under a catalogue number of the form Knnn. We sincerely thank Edward Brisse, Fred Lang, Wilson Stothers and Paul Yiu for their friendly support and help. Chapter 1 Preliminaries and definitions 1.1 Notations We will denote by the cubic curve with barycentric equation • K F (x,y,z) = 0 where F is a third degree homogeneous polynomial in x,y,z. Its partial derivatives ∂F ∂2F will be noted F ′ for and F ′′ for when no confusion is possible. x ∂x xy ∂x∂y Any cubic with three real distinct asymptotes making 60◦ angles with one another • will be called an equilateral cubic or a 60.
    [Show full text]
  • Generalized Gergonne and Nagel Points
    Generalized Gergonne and Nagel Points B. Odehnal June 16, 2009 Abstract In this paper we show that the Gergonne point G of a triangle ∆ in the Euclidean plane can in fact be seen from the more general point of view, i.e., from the viewpoint of projective geometry. So it turns out that there are up to four Gergonne points associated with ∆. The Gergonne and Nagel point are isotomic conjugates of each other and thus we find up to four Nagel points associated with a generic triangle. We reformulate the problems in a more general setting and illustrate the different appearances of Gergonne points in different affine geometries. Darboux’s cubic can also be found in the more general setting and finally a projective version of Feuerbach’s circle appears. Mathematics Subject Classification (2000): 51M04, 51M05, 51B20 Key Words: triangle, incenter, excenters, Gergonne point, Brianchon’s theorem, Nagel point, isotomic conjugate, Darboux’s cubic, Feuerbach’s nine point circle. 1 Introduction Assume ∆ = {A,B,C} is a triangle with vertices A, B, and C and the incircle i. Gergonne’s point is the locus of concurrency of the three cevians connecting the contact points of i and ∆ with the opposite vertices. The Gergonne point, which can also be labelled with X7 according to [10, 11] has frequently attracted mathematician’s interest. Some generalizations have been given. So, for example, one can replace the incircle with a concentric 1 circle and replace the contact points of the incircle and the sides of ∆ with their reflections with regard to the incenter as done in [2].
    [Show full text]
  • 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.
    [Show full text]
  • A Note on the Miquel Points
    International Journal of Computer Discovered Mathematics (IJCDM) ISSN 2367-7775 c IJCDM September 2016, Volume 1, No.3, pp.45-49. Received 10 September 2016. Published on-line 20 September 2016 web: http://www.journal-1.eu/ c The Author(s) This article is published with open access1. Computer Discovered Mathematics: A Note on the Miquel Points Sava Grozdeva, Hiroshi Okumurab and Deko Dekovc 2 a VUZF University of Finance, Business and Entrepreneurship, Gusla Street 1, 1618 Sofia, Bulgaria e-mail: [email protected] b Department of Mathematics, Yamato University, Osaka, Japan e-mail: [email protected] cZahari Knjazheski 81, 6000 Stara Zagora, Bulgaria e-mail: [email protected] web: http://www.ddekov.eu/ Abstract. By using the computer program “Discoverer”, we give theorems about Miquel associate points. Abstract. Keywords. Miquel associate point, triangle geometry, remarkable point, computer-discovered mathematics, Euclidean geometry, Discoverer. Mathematics Subject Classification (2010). 51-04, 68T01, 68T99. 1. Introduction The computer program “Discoverer”, created by Grozdev and Dekov, with the collaboration by Professor Hiroshi Okumura, is the first computer program, able easily to discover new theorems in mathematics, and possibly, the first computer program, able easily to discover new knowledge in science. In this paper, by using the “Discoverer”, we investigate the Miquel points. The following theorem is known as the Miquel theorem: Theorem 1.1. If points A1;B1 and C1 are chosen on the sides BC; CA and AB of triangle ABC, then the circumcircles of triangles AB1C1; BC1A1 and CA1B1 have a point in common. 1This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
    [Show full text]
  • Nagel , Speiker, Napoleon, Torricelli
    Nagel , Speiker, Napoleon, Torricelli MA 341 – Topics in Geometry Lecture 17 Centroid The point of concurrency of the three medians. 07-Oct-2011 MA 341 2 Circumcenter Point of concurrency of the three perpendicular bisectors. 07-Oct-2011 MA 341 3 Orthocenter Point of concurrency of the three altitudes. 07-Oct-2011 MA 341 4 Incenter Point of concurrency of the three angle bisectors. 07-Oct-2011 MA 341 5 Symmedian Point Point of concurrency of the three symmedians. 07-Oct-2011 MA 341 6 Gergonne Point Point of concurrency of the three segments from vertices to intangency points. 07-Oct-2011 MA 341 7 Spieker Point The Spieker point is the incenter of the medial triangle. 07-Oct-2011 MA 341 8 Nine Point Circle Center The 9 point circle center is midpoint of the Euler segment. 07-Oct-2011 MA 341 9 Mittenpunkt Point The mittenpunkt of ΔABC is the symmedian point of the excentral triangle 05-Oct-2011 MA 341 001 10 Nagel Point Nagel point = point of concurrency of cevians to points of tangency of excircles 05-Oct-2011 MA 341 001 11 Nagel Point 05-Oct-2011 MA 341 001 12 The Nagel Point Ea has the unique property of being the point on the perimeter that is exactly half way around the triangle from A. = AB+BEaa AC+CE =s Then =- =- =- =- BEaa s AB s c and CE s AC s b BE sc- a = - CEa s b 07-Oct-2011 MA 341 13 The Nagel Point Likewise CE sa- b = - AEb s c AE sb- c = - BEc s a Apply Ceva’s Theorem AE BE CE sbscsa--- cab=´´=1 --- BEcab CE AE s a s b s c 07-Oct-2011 MA 341 14 The Nagel Segment 1.
    [Show full text]
  • Volume 6 (2006) 1–16
    FORUM GEOMETRICORUM A Journal on Classical Euclidean Geometry and Related Areas published by Department of Mathematical Sciences Florida Atlantic University b bbb FORUM GEOM Volume 6 2006 http://forumgeom.fau.edu ISSN 1534-1178 Editorial Board Advisors: John H. Conway Princeton, New Jersey, USA Julio Gonzalez Cabillon Montevideo, Uruguay Richard Guy Calgary, Alberta, Canada Clark Kimberling Evansville, Indiana, USA Kee Yuen Lam Vancouver, British Columbia, Canada Tsit Yuen Lam Berkeley, California, USA Fred Richman Boca Raton, Florida, USA Editor-in-chief: Paul Yiu Boca Raton, Florida, USA Editors: Clayton Dodge Orono, Maine, USA Roland Eddy St. John’s, Newfoundland, Canada Jean-Pierre Ehrmann Paris, France Chris Fisher Regina, Saskatchewan, Canada Rudolf Fritsch Munich, Germany Bernard Gibert St Etiene, France Antreas P. Hatzipolakis Athens, Greece Michael Lambrou Crete, Greece Floor van Lamoen Goes, Netherlands Fred Pui Fai Leung Singapore, Singapore Daniel B. Shapiro Columbus, Ohio, USA Steve Sigur Atlanta, Georgia, USA Man Keung Siu Hong Kong, China Peter Woo La Mirada, California, USA Technical Editors: Yuandan Lin Boca Raton, Florida, USA Aaron Meyerowitz Boca Raton, Florida, USA Xiao-Dong Zhang Boca Raton, Florida, USA Consultants: Frederick Hoffman Boca Raton, Floirda, USA Stephen Locke Boca Raton, Florida, USA Heinrich Niederhausen Boca Raton, Florida, USA Table of Contents Khoa Lu Nguyen and Juan Carlos Salazar, On the mixtilinear incircles and excircles,1 Juan Rodr´ıguez, Paula Manuel and Paulo Semi˜ao, A conic associated with the Euler line,17 Charles Thas, A note on the Droz-Farny theorem,25 Paris Pamfilos, The cyclic complex of a cyclic quadrilateral,29 Bernard Gibert, Isocubics with concurrent normals,47 Mowaffaq Hajja and Margarita Spirova, A characterization of the centroid using June Lester’s shape function,53 Christopher J.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Geometry of the Triangle
    Geometry of the Triangle Paul Yiu 2016 Department of Mathematics Florida Atlantic University A b c B a C August 18, 2016 Contents 1 Some basic notions and fundamental theorems 101 1.1 Menelaus and Ceva theorems ........................101 1.2 Harmonic conjugates . ........................103 1.3 Directed angles ...............................103 1.4 The power of a point with respect to a circle ................106 1.4.1 Inversion formulas . ........................107 1.5 The 6 concyclic points theorem . ....................108 2 Barycentric coordinates 113 2.1 Barycentric coordinates on a line . ....................113 2.1.1 Absolute barycentric coordinates with reference to a segment . 113 2.1.2 The circle of Apollonius . ....................114 2.1.3 The centers of similitude of two circles . ............115 2.2 Absolute barycentric coordinates . ....................120 2.2.1 Homotheties . ........................122 2.2.2 Superior and inferior . ........................122 2.3 Homogeneous barycentric coordinates . ................122 2.3.1 Euler line and the nine-point circle . ................124 2.4 Barycentric coordinates as areal coordinates ................126 2.4.1 The circumcenter O ..........................126 2.4.2 The incenter and excenters . ....................127 2.5 The area formula ...............................128 2.5.1 Conway’s notation . ........................129 2.5.2 Conway’s formula . ........................130 2.6 Triangles bounded by lines parallel to the sidelines . ............131 2.6.1 The symmedian point . ........................132 2.6.2 The first Lemoine circle . ....................133 3 Straight lines 135 3.1 The two-point form . ........................135 3.1.1 Cevian and anticevian triangles of a point . ............136 3.2 Infinite points and parallel lines . ....................138 3.2.1 The infinite point of a line .
    [Show full text]
  • 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.
    [Show full text]