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Volume 14 2014 FORUM GEOMETRICORUM A Journal on Classical Euclidean Geometry and Related Areas published by Department of Mathematical Sciences Florida Atlantic University FORUM GEOM Volume 14 2014 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: Nikolaos Dergiades Thessaloniki, Greece 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 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 Martin Josefsson, Angle and circle characterizations of tangential quadrilaterals,1 Paris Pamfilos, The associated harmonic quadrilateral,15 Gregoire´ Nicollier, Dynamics of the nested triangles formed by the tops of the perpendicular bisectors,31 J. Marshall Unger, Kitta’s double-locked problem,43 Marie-Nicole Gras, Distances between the circumcenter of the extouch triangle and the classical centers of a triangle,51 Sander´ Kiss and Paul Yiu, The touchpoints triangles and the Feuerbach hyperbolas, 63 Benedetto Scimemi, Semi-similar complete quadrangles,87 Jose´ L. Ram´ırez, Inversions in an ellipse, 107 Bryan Brzycki, On a geometric locus in taxicab geometry, 117 Dao Thanh Oai, A simple proof of Gibert’s generalization of the Lester circle theorem, 123 Cristinel Mortici, A note on the Fermat-Torricelli point of a class of polygons, 127 Martin Josefsson, Properties of equidiagonal quadrilaterals, 129 Michal Rol´ınek and Le Anh Dung, The Miquel points, pseudocircumcenter, and Euler-Poncelet point of a complete quadrilateral , 145 Emmanuel Antonio Jose´ Garc´ıa, A note on reflections, 155 Nikolaos Dergiades, Antirhombi, 163 Bernard Gibert, Asymptotic directions of pivotal isocubics, 173 Bernard Gibert, The Cevian Simson transformation, 191 Dao Thanh Oai, Two pairs of Archimedean circles in the arbelos, 201 Gotthard Weise, On some triads of homothetic triangles, 203 Manfred Evers, Symbolic substitution has a geometric meaning, 217 Michael de Villiers, Quasi-circumcenters and a generalization of the quasi-Euler line to a hexagon, 233 Surajit Dutta, A simple property of isosceles triangles with applications, 237 Hiroshi Okumura, A note on Haga’s theorems in paper folding, 241 Nikolaos Dergiades, Dao’s theorem on six circumcenters associated with a cyclic hexagon, 243 Tran Quang Hung, Two tangent circles from jigsawing quadrangle, 247 Tran Quang Hung, Two more pairs of Archimedean circles in the arbelos, 249 Floor van Lamoen, A special point in the arbelos leading to a pair of Archimedean circles, 253 Paul Yiu, Three constructions of Archimedean circles in an arbelos, 255 Telv Cohl, A purely synthetic proof of Dao’s theorem on six circumcenters associated with a cyclic hexagon, 261 Jesus Torres, The triangle of reflections, 265 Paris Pamfilos, A gallery of conics with five elements, 295 Shao-Cheng Liu, On two triads of triangles associated with the perpendicular bisectors of the sides of a triangle, 349 Hiroshi Okumura, Archimedean circles related to the Schoch line, 369 Thierry Gensane and Pascal Honvault, Optimal packings of two ellipses in a square, 371 Martin Josefsson, The diagonal point triangle revisited, 381 Francisco Javier Garc´ıa Capitan,´ A simple construction of an inconic, 387 Albrecht Hess, On a circle containing the incenters of tangential quadrilaterals, 389 Mihaly´ Bencze and Ovidiu T. Pop, Congruent contiguous excircles, 397 Nguyen Thanh Dung, Some circles associated with the Feuerbach points, 403 Nikolaos Dergiades, Generalized Archimedean arbelos twins, 409 Author Index, 419 Forum Geometricorum b Volume 14 (2014) 1–13. b b FORUM GEOM ISSN 1534-1178 Angle and Circle Characterizations of Tangential Quadrilaterals Martin Josefsson Abstract. We prove five necessary and sufficient conditions for a convex quadri- lateral to have an incircle that concerns angles or circles. 1. Introduction A tangential quadrilateral is a convex quadrilateral with an incircle, i.e., a circle inside the quadrilateral that is tangent to all four sides. In [4] and [5] we reviewed and proved a total of 20 different necessary and sufficient conditions for a convex quadrilateral to be tangential. Of these there were 14 dealing with different dis- tances (sides, line segments, radii, altitudes), four were about circles (excluding their radii), and only two were about angles. In this paper we will prove five more such characterizations concerning angles and circles. First we review two that can be found elsewhere. A characterization involving the four angles and all four sides of a quadrilateral appeared as part of a proof of an inverse altitude characterization of tangential quadrilaterals in [6, p.115]. According to it, a convex quadrilateral ABCD with sides a = AB, b = BC, c = CD and d = DA is tangential if and only if a sin A sin B + c sin C sin D = b sin B sin C + d sin D sin A. In the extensive monograph [9, p.133] on quadrilateral geometry, the following characterization is attributed to Simionescu. A convex quadrilateral is tangential if and only if its consecutive sides a, b, c, d and diagonals p, q satisfies ac bd = pq cos θ | − | where θ is the acute angle between the diagonals. The proof is a simple application of the quite well known identity 2pq cos θ = b2 + d2 a2 c2 that holds in all − − convex quadrilaterals. Rewriting it as 2pq cos θ = (b + d)2 (a + c)2 + 2(ac bd) , − − we see that Simionescu’s theorem is equivalent to Pitot’s theorem a + c = b + d for tangential quadrilaterals. In Theorem 2 we will prove another characterization for the angle between the diagonals, but it only involves four different distances instead of six. Publication Date: January 23, 2014. Communicating Editor: Paul Yiu. 2 M. Josefsson 2. Angle characterizations of tangential quadrilaterals It is well known that a convex quadrilateral has an incircle if and only if the four angle bisectors of the internal vertex angles are concurrent. If this point exist, it is the incenter. Here we shall prove a necessary and sufficient condition for an incircle regarding the intersection of two opposite angle bisectors which characterize the incenter in terms of two angles in two different ways. To prove that one of these equalities holds in a tangential quadrilateral (the direct theorem) was a problem in [1, p.67]. Theorem 1. A convex quadrilateral ABCD is tangential if and only if ∠AIB + ∠CID = π = ∠AID + ∠BIC where I is the intersection of the angle bisectors at A and C. Proof. ( ) In a tangential quadrilateral the four angle bisectors intersect at the incenter.⇒ Using the sum of angles in a triangle and a quadrilateral, we have A B C D 2π ∠AIB + ∠CID = π + + π + = 2π = π. − 2 2 − 2 2 − 2 The second equality can be proved in the same way, or we can use that the four angles in the theorem make one full circle, so ∠AID + ∠BIC = 2π π = π. − D b C b I b b D′′ b b b A B D′ Figure 1. Construction of the points D′ and D′′ ( ) In a convex quadrilateral where I is the intersection of the angle bisectors at A⇐and C, and the equality ∠AIB + ∠CID = ∠AID + ∠BIC (1) holds, assume without loss of generality that AB > AD and BC > CD. 1 Con- struct points D′ and D′′ on AB and BC respectively such that AD′ = AD and CD′′ = CD (see Figure 1). Then triangles AID′ and AID are congruent, and so are triangles CID′′ and CID. Thus ID′ = ID = ID′′. These two pairs 1If instead there is equality in one of these inequalities, then it’s easy to see that the quadrilateral is a kite. It’s well known that kites have an incircle. Angle and circle characterizations of tangential quadrilaterals 3 of congruent triangles and (1) yields that ∠BID′ = ∠BID′′, so triangles BID′ and BID′′ are congruent. Thus BD′ = BD′′. Together with AD′ = AD and CD′′ = CD, we get ′ ′ ′′ ′′ AD + D B + CD = AD + BD + D C AB + CD = AD + BC. ⇒ Then ABCD is a tangential quadrilateral according to Pitot’s theorem. The idea for the proof of the converse comes from [8], where Goutham used this method to prove the converse of a related characterization of tangential quadrilat- erals concerning areas. That characterization states that if I is the intersection of the angle bisectors at A and C in a convex quadrilateral ABCD, then it has an incircle if and only if SAIB + SCID = SAID + SBIC , where SXYZ stands for the area of triangle XYZ. According to [9, p.134], this theorem is due to V. Pop and I. Gavrea. In [6, pp.117–118] a similar characteriza- tion concerning the same four areas was proved, but it also includes the four sides. It states that ABCD is a tangential quadrilateral if and only if c SAIB + a SCID = b SAID + d SBIC , · · · · where a = AB, b = BC, c = CD and d = DA. The next characterization is about the angle between the diagonals. We will assume we know the lengths of the four parts that the intersection of the diagonals divide them into.
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