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Volume 8 (2008) 1–5 FORUM GEOMETRICORUM A Journal on Classical Euclidean Geometry and Related Areas published by Department of Mathematical Sciences Florida Atlantic University b b b FORUM GEOM Volume 8 2008 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 Steve Sigur Atlanta, Georgia, 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 Jean-Pierre Ehrmann, An affine variant of a Steinhaus problem,1 Quang Tuan Bui, Two triads of congruent circles from reflections,7 George Baloglou and Michel Helfgott, Angles, area, and perimeter caught in a cubic, 13 Panagiotis T. Krasopoulos, Kronecker theorem and a sequence of triangles, 27 Mowaffaq Hajja, A short trigonometric proof of the Steiner-Lehmus theorem, 39 Cosmin Pohoata, On the Parry reflection point, 43 Floor van Lamoen and Paul Yiu, Construction of Malfatti squares, 49 Kurt Hofstetter, A simple ruler and rusty compass construction of the regular pentagon, 61 Yaroslav Bezverkhnyev, Haruki’s lemma and a related locus problem, 63 Wei-Dong Jiang, An inequality involving the angle bisectors and an interior point of a triangle, 73 Bernard Gibert, Cubics related to coaxial circles, 77 Cosmin Pohoata, A short proof of Lemoine’s theorem, 97 Francisco Javier Garc´ıa Capit´an, Means as chords, 99 Mowaffaq Hajja, A condition for a circumscriptible quadrilateral to be cyclic, 103 Nicolas Bedaride, Periodic billiard trajectories in polyhedra, 107 Maria Flavia Mammana, Biagio Micale, and Mario Pennisi, On the centroids of polygons and polyhedra, 121 Sadi Abu-Saymeh, Mowaffaq Hajja, and Hassan Ali ShahAli, Another variation of the Steiner-Lehmus theme, 131 Yaroslav Bezverkhnyev, Haruki’s lemma for conics, 141 Kurt Hofstetter, A simple compass-only construction of the regular pentagon, 147 Quang Tuan Bui, Two more Powerian pairs in the arbelos, 149 Mikl´os Hoffmann and Sonja Gorjanc, On the generalized Gergonne point and beyond, 151 Mowaffaq Hajja, Stronger forms of the Steiner-Lehmus theorem, 157 Yu-Dong Wu, A new proof of a weighted Erdos-Mordell˝ type inequality, 163 Michel Bataille, Another compass-only construction of the golden section and of the regular pentagon, 167 Giovanni Lucca, Some identities arising from inversion of Pappus chains in an arbelos, 171 Clark Kimberling, Second-Degree Involutory Symbolic Substitutions, 175 Jan Vonk, On the Nagel line and a prolific polar triangle, 183 Victor Oxman, A purely geometric proof of the uniqueness of a triangle with prescribed angle bisectors, 197 Eisso J. Atzema, An elementary proof of a theorem by Emelyanov, 201 Shao-Cheng Liu, A generalization of Thebault’s´ theorem on the concurrency of three Euler lines, 205 Author Index, 209 Forum Geometricorum b Volume 8 (2008) 1–5. b b FORUM GEOM ISSN 1534-1178 On an Affine Variant of a Steinhaus Problem Jean-Pierre Ehrmann Abstract. Given a triangle ABC and three positive real numbers u, v, w, we prove that there exists a unique point P in the interior of the triangle, with ce- vian triangle PaPbPc, such that the areas of the three quadrilaterals P PbAPc, P PcBPa , P PaCPb are in the ratio u : v : w. We locate P as an intersection of three hyperbolas. In this note we study a variation of the theme of [2], a generalization of a prob- lem initiated by H. Steinhaus on partition of a triangle (see [1]). Given a triangle ABC with interior T , and a point P ∈ T with cevian triangle PaPbPc, we de- note by ∆A(P ), ∆B(P ), ∆C (P ) the areas of the oriented quadrilaterals P PbAPc, P PcBPa, P PaCPb. In this note we prove that given three arbitrary positive real numbers u, v, w, there exists a unique point P ∈ T such that ∆A(P ):∆B(P ):∆C (P )= u : v : w. To this end, we define f(P )=∆A(P ):∆B(P ):∆C (P ). This is the point of T such that ∆[BCf(P )]=∆A(P ), ∆[CAf(P )]=∆B(P ), ∆[ABf(P )]=∆C (P ). Lemma 1. If P has homogeneous barycentric coordinates x : y : z with reference to triangle ABC, then (y + z)(2x + y + z) (z + x)(2y + z + x) (x + y)(x + y + 2z) f(P )= : : . x y z Proof. If P = x : y : z, we have −−→ −−→ −→ −→ −−→ yAB −→ y AB + z AC −−→ zAC APc = , AP = , AP = , x + y x + y + z b x + z so that y z 1 1 ∆a(P )=∆(APcP )+∆(AP Pb)= ( + ) ∆(ABC). x + y + z x + y x + z Publication Date: January 7, 2008. Communicating Editor: Paul Yiu. 2 J.-P. Ehrmann By cyclic permutations of x, y, z, we get the values of ∆B(P ) and ∆C (P ), and the result follows. We shall prove that f : T → T is a bijection. We adopt the following nota- tions. (i) Ga, Gb, Gc are the vertices of the anticomplementary triangle. They are the images A, B, C under the homothety h(G, −2), G being the centroid of ABC. ∗ ∗ (ii) P denotes the isotomic conjugate of P with respect to ABC. Its traces Pa , ∗ ∗ Pb , Pc on the sidelines of ABC are the reflections of Pa, Pb, Pc with respect to the midpoint of the corresponding side. (iii) [L]∞ denotes the infinite point of a line L. ∗ Proposition 2. Let P = x : y : z and U = u : v : w. The lines GaP and Pa U ∗ are parallel if and only if P lies on the hyperbola Ha,U through A, Ga, Ua , the ∗ ∗ reflection of Ub in C and the reflection of Uc in B. ∗ Proof. As Pa = 0 : z : y and [GaP ]∞ = −(2x + y + z) : z + x : x + y, the lines ∗ GaP and Pa U are parallel if and only if ∗ ha,U (P ) := det([GaP ]∞, Pa , U) −(2x + y + z) z + x x + y = 0 z y u v w = x((u + v)y − (w + u)z) + (x + y + z)(vy − wz) = 0. It is clear that ha,U (P ) = 0 defines a conic Ha,U through A =1:0:0, and the infinite points of the lines x = 0 and (u + v)y − (w + u)z = 0. These are the lines BC and GaU. It is also easy to check that it contains the points Ga = −1:1:1, ∗ Ua = 0 : w : v, and ∗ Ubc := − w : 0 : u + 2w, ∗ Ucb := − v : u + 2v : 0. ∗ ∗ These latter two are respectively the reflections of Ub in C and Uc in B. The conic H ∗ ∗ a,U is a hyperbola since the four points A, Ga, Ubc and Ucb do not fall on two lines. By cyclic permutations of coordinates, we obtain two hyperbolae Hb,U and Hc,U defined by ∗ hb,U (P ) := det([GbP ]∞, Pb , U) = 0, ∗ hc,U (P ) := det([GcP ]∞, Pc , U) = 0. It is easy to check that if U = f(P ), then ha,U (P )= hb,U (P )= hc,U (P ) = 0. From this we obtain a very easy construction of the point f(P ). On an affine variant of a Steinhaus problem 3 ∗ ∗ Corollary 3. The point f(P ) is the intersection of the lines through Pa , Pb and ∗ Pc parallel to GaP , GbP , GcP respectively. See Figure 1. Gc A Gb ∗ Pc ∗ Pb P P ∗ f(P ) ∗ B Pa C Ga Figure 1. ∗ ∗ ∗ Proof. The lines GaP , GbP , GcP are parallel to Pa f(P ), Pb f(P ), Pc f(P ) re- spectively. Remarks. (1) Ha,U degenerates if and only if v = w, i.e., when U lies on the median AG. In this case, Ha,U is the union of the median AG and of a line parallel to BC. (2) P , P ∗, f(P ) are collinear. (3) As ha,U (P )+ hb,U (P )+ hc,U (P ) = 0, the three hyperbolae Ha,U , Hb,U , Hc,U are members of a pencil of conics. If U ∈ T , the points P for which f(P )= U are their common points lying in T . Lemma 4. If U ∈ T , Ha,U and Hb,U have a real common point in T and a real common point in TA, reflection in A of the open angular sector bounded by the half lines AB and AC. Proof. Using the fact that Ha,U passes through [BC]∞, we can cut Ha,U by lines parallel to BC to get a rational parametrization of Ha,U . More precisely, let Bt and Ct be the images of B and C under the homothety h(A, 1 − t). The point (1 − µ)Bt + µCt = t : (1 − µ)(1 − t) : µ(1 − t) lies on Ha,U if and only if v + t(u + v) µ = µt = . v + w + t(2u + v + w) 4 J.-P. Ehrmann Let P (t) = (1 − µt)Bt + µtCt. It has homogeneous barycentric coordinates t((v + w)+ t(2u + v + w)) : (1 − t)(w + t(w + u)) : (1 − t)(v + t(u + v)).
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