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Volume 18 2018 FORUM GEOMETRICORUM A Journal on Classical Euclidean Geometry and Related Areas published by Department of Mathematical Sciences Florida Atlantic University FORUM GEOM Volume 18 2018 http://forumgeom.fau.edu ISSN 1534-1188 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: Eisso J. Atzema Orono, Maine, USA Nikolaos Dergiades Thessaloniki, Greece 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 Man Keung Siu Hong Kong, China Peter Woo La Mirada, California, USA Li Zhou Winter Haven, Florida, 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 Stefan Liebscher and Dierck-E. Liebscher, The relativity of conics and circles,1 Carl Eberhart, Revisiting the quadrisection problem of Jacob Bernoulli,7 C. E. Garza-Hume, Maricarmen C. Jorge, and Arturo Olvera, Areas and shapes of planar irregular polygons,17 √ Samuel G. Moreno and Esther M. Garc´ıa–Caballero, Irrationality of 2:Yet another visual proof,37 Manfred Pietsch, Two hinged regular n-sided polygons, 39–42 Hiroshi Okumura, A remark on the arbelos and the regular star polygon,43 Lubomir P. Markov, Revisiting the infinite surface area of Gabriel’s horn,45 Giovanni Lucca, Integer sequences and circle chains inside a circular segment,47 Michel Bataille, On the extrema of some distance ratios,57 Apostolos Hadjidimos, Twins of Hofstadter elements,63 Li Zhou, Primitive Heronian triangles with integer inradius and exradii,71 Francisco Javier Garc´ıa Capitan,´ A family of triangles for which two specific triangle centers have the same coordinates,79 Robert Bosch, A new proof of Erdos-Mordell˝ inequality,83 Paris Pamfilos, Parabola conjugate to rectangular hyperbola,87 Gerasimos T. Soldatos, A toroidal approach to the doubling of the cube,93 Hiroshi Okumura and Saburou Saitoh, Remarks for the twin circles of Archimedes in a skewed arbelos,99 (Retracted) Sandor´ Nagydobai Kiss, On the cyclic quadrilaterals with the same Varignon parallelogram, 103 Gabor´ Gevay,´ An extension of Miquel’s six-circles theorem, 115 Nicholas D. Brubaker, Jasmine Camero, Oscar Rocha Rocha, Roberto Soto, and Bogdan D. Suceava,˘ A curvature invariant inspired by Leonhard Euler’s inequality R ≥ 2r, 119 Michel Bataille, Constructing a triangle from two vertices and the symmedian point, 129 Eugen J. Ionas¸cu, The “circle” of Apollonius in Hyperbolic Geometry, 135 Mark Shattuck, A geometric inequality for cyclic quadrilaterals, 141 Hiroshi Okumura and Saburou Saitoh, Harmonic mean and division by zero, 155 Paris Pamfilos, Rectangles circumscribing a quadrangle, 161 Mihaly´ Bencze and Marius Dragan,˘ The Blundon theorem in an acute triangle and some consequences, 185 Purevsuren Damba and Uuganbaatar Ninjbat, Side disks of a spherical great polygon, 195 Nikolaos Dergiades, Parallelograms inscribed in convex quadrangles, 203 Mohammad K. Azarian, “A Study of Risala¯ al-Watar wa’l Jaib” (The Treatise on the Chord and Sine): Revisited, 219 Blas Herrera, Algebraic equations of all involucre conics in the configuration of the c-inscribed equilateral triangles of a triangle, 223 Tran Quang Hung, A construction of the golden ratio in an arbitrary triangle, 239 Martina Stˇ epˇ anov´ a,´ New Constructions of Triangle from α, b − c, tA, 245 Robert Bosch, A new proof of Pitot theorem by AM-GM inequality, 251 John Donnelly, A model of continuous plane geometry that is nowhere geodesic, 255 John Donnelly, A model of nowhere geodesic plane geometry in which the triangle inequality fails everywhere, 275 Omid Ali Shahni Karamzadeh, Is the mystery of Morley’s trisector theorem resolved ? 297 Floor van Lamoen, A synthetic proof of the equality of iterated Kiepert triangles K(φ, ψ)=K(ψ, φ), 307 S¸ahlar Meherrem, Gizem Gunel¨ Ac¸ıksoz,¨ Serenay S¸en, Zeynep Sezer, and Gunes¨ ¸ Bas¸kes, Geometric inequalities in pedal quadrilaterals, 309 Paris Pamfilos, Self-pivoting convex quadrangles, 321 Floor van Lamoen, Orthopoles and variable flanks, 349 Nikolaos Dergiades and Tran Quang Hung, Simple proofs of Feuerbach’s Theorem and Emelyanov’s Theorem, 353 Roger C. Alperin, Pedals of the Poncelet pencil and Fontene´ points, 361 Francisco Javier Garc´ıa Capitan,´ Circumconics with asymptotes making a given angle, 367 Yong Zhang, Junyao Peng, and Jiamin Wang, Integral triangles and trapezoids pairs with a common area and a common perimeter, 371 Giuseppina Anatriello, Francesco Laudano, and Giovanni Vincenzi, Pairs of congruent -like quadrilaterals that are not congruent, 381 Gabor´ Gevay,´ A remarkable theorem on eight circles, 401 Hiroshi Okumura, An analogue to Pappus chain theorem with division by zero, 409 Rogerio´ Cesar´ dos Santos, Polygons on the sides of an octagon, 413 Stefan Liebscher and Dierck E.-Liebscher, The love for the three conics, 419 Gerasimos T. Soldatos, A polynomial approach to the “Bloom” of Thymaridas and the Apollonius’ circle, 431 Author index, 435 Forum Geometricorum b Volume 18 (2018) 1–6. b b FORUM GEOM ISSN 1534-1178 The Relativity of Conics and Circles Stefan Liebscher and Dierck-E. Liebscher Abstract. Foci are defined for a pair of conics. They are the six vertices of the quadrilateral of common tangents. To be circle is a derived property of a pair of conics, too. 1. Introduction The beloved properties of conics, which we start to discover at school, manifest themselves as projective relations between two conics, as soon as we use the em- bedding in the non-Euclidean Cayley-Klein geometries. Non-Euclidean geometry enables us to find real constructions of objects which otherwise lie in infinity or are imaginary. Our school-level understanding of geometry is a play with ruler and compass on the Euclidean plane. Conics constitute the last topic to be tackled in this manner and the first that leads beyond its limits. The most prominent property of conics is the existence of foci. In non-Euclidean geometry, the scale on the drawing plane is not constant. So, how can we define motions and determine congruency? We start by generating the group of motions by defining reflections [1]. Reflections define perpendiculars, and in Cayley-Klein geometries, a conic provides the reflections simply by requiring its own invariance [1]. Reflection of tangents to this absolute conic yield other tangents again, and the reflection of a point is the intersection of the reflection of its two tangents to the absolute conic. Two lines are perpendicular if one is reflected by the other onto itself. Such lines contain the others pole. The pole becomes the common intersection of all perpendiculars of its polar. Geometries can now be classified by their absolute conics. In Euclidean geom- etry, this conic is a double line at infinity with complex fixed points, so it is not prominent. The line at infinity contains the poles of all lines and is the polar of all points. In Minkowski geometry, the fixed points are real, and become prominent in their role as directions of the world lines of light signals [2]. In Galilei geometry, these two fixed points coincide [5]. The Beltrami-Klein model of the hyperbolic plane uses a regular conic. To adapt to our Euclidean habits, it has the form of a circle. As we shall see, a conic acquires the properties of a circle when declared absolute. Publication Date: January 16, 2018. Communicating Editor: Paul Yiu. 2 S. Liebscher and D-E. Liebscher Left: Given a set of conics with two common tangents. The poles of any line passing the vertex of the tangents are concurrent with this vertex. Right: Two conics define four common tangents and generate a pencil of conics. Any two conics of the pencil have the same quadrilateral of common tangents with the same six vertices and three diagonals (dashed lines). The vertices are foci by this pole-collinearity property. If any conic of the pencil is declared absolute, the foci obtain the familiar metric properties of a focus, too. Figure 1. Confocal conics for real common tangents 2. Foci as properties of pairs of conics The focus of a conic is characterized by a particular property of its pencil of rays: The pole (with respect to the conic considered) of any line through the focus lies on the perpendicular to the line in the focus. In other words, the poles of a focal ray with respect to the two conics (the considered and the absolute) are collinear with the focus. This statement does not refer to the task of the absolute conic to define perpendicularity. It solely uses the pole-polar relation of two conics. Foci thus become properties of pairs of conics instead of a single conic and the metric. Let us consider a tangent from the focus to one of the conics, so that its pole is the contact point. The pole by the other conic can be collinear with this contact point and the focus only when it lies on the tangent, too. That is, the line is tangent to both conics. A focus must be an intersection of tangents common to the conics, whether real or not. Consequently, the intersections of the common tangents are the foci (Fig. 1, left). The foci of a pair of conics are the vertices of the quadrilateral of common tangents.
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