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 11 2011 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 Alain Levelut, A note on the Hervey point of a complete quadrilateral 1 Arie Bialostocki and Dora Bialostocki, The incenter and an excenter as solutions to an extremal problem, 9 Jan Vonk and J. Chris Fisher, Translation of Fuhrmann’s “Sur un nouveau cercle associe´ a` un triangle”, 13 Paris Pamfilos, Triangles with given incircle and centroid, 27 Jo Niemeyer, A simple construction of the golden section, 53 Michel Bataille, Another simple construction of the golden section, 55 Michel Bataille, On the foci of circumparabolas, 57 Martin Josefsson, More characterizations of tangential quadrilaterals, 65 Peter J. C. Moses and Clark Kimberling, Perspective isoconjugate triangle pairs, Hofstadter pairs, and crosssums on the nine-point circle, 83 Darren C. Ong, On a theorem of intersecting conics, 95 Maria Flavia Mammana, Biagio Micale, and Mario Pennisi, The Droz-Farny circles of a convex quadrilateral, 109 Roger C. Alperin, Solving Euler’s triangle problems with Poncelet’s pencil, 121 Nguyen Minh Ha and Bui Viet Loc, More on the extension of Fermat’s problem, 131 Hiroshi Okumura, More on twin circles of the skewed arbelos, 139 Luis Gonz´alez, On a triad of circles tangent to the circumcircle and the sides at their midpoints, 145 Martin Josefsson, The area of a bicentric quadrilateral, 155 Martin Josefsson, When is a tangential quadrilateral a kite ? 165 Toufik Mansour and Mark Shattuck, On a certain cubic geometric inequality, 175 Joel C. Langer and David A. Singer, The lemniscatic chessboard, 183 Floor van Lamoen, A spatial view of the second Lemoine circle, 201 Michael J. G. Scheer, A simple vector proof of Feuerbach’s theorem, 205 Larry Hoehn, A new formula concerning the diagonals and sides of a quadrilateral, 211 Martin Josefsson, The area of the diagonal point triangle, 213 Francisco Javier Garc´ıa Capit´an, Collinearity of the first trisection points of cevian segments, 217 Mihai Cipu, Cyclic quadrilaterals associated with squares, 223 William N. Franzsen, The distance from the incenter to the Euler line, 231 Paul Yiu, Rational Steiner porism, 237 Quang Tuan Bui, Golden sections in a regular hexagon, 251 Nikolaos Dergiades and Paul Yiu, The golden section with a collapsible compass only, 255 Jean-Pierre Ehrmann, Francisco Javier Garc´ıa Capit´an, and Alexei Myakishev, Construction of circles through intercepts to paralells to cevians, 261 Nikolas Dergiades, Francisco Javier Garc´ıa Capit´an, and Sung Hyun Lim, On six circumcenters and their concyclicity, 269 Author Index, 277 Forum Geometricorum b Volume 11 (2011) 1–7. b b FORUM GEOM ISSN 1534-1178 A Note on the Hervey Point of a Complete Quadrilateral Alain Levelut Abstract. Using the extension of the Sylvester relation to four concyclic points we define the Hervey point of a complete quadrilateral and show that it is the center of a circle congruent to the Miquel circle and that it is the point of concur- rence of eight remarkable lines of the complete quadrilateral. We also show that the Hervey point and the Morley point coincide for a particular type of complete quadrilaterals. 1. Introduction In the present paper we report on several properties of a remarkable point of a complete quadrilateral, the so-called Hervey point. 1 This originated from a prob- lem proposed and solved by F. R. J. Hervey [2] on the concurrence of the four lines drawn through the centers of the nine point circles perpendicular to the Euler lines of the four associated triangles. Here, we use a different approach which seems more efficient than the usual one. We first define the Hervey point by the means of a Sylvester type relation, and, only after that, we look for its properties. Following this way, several interesting results are easily obtained: the Hervey point is the cen- ter of a circle congruent to the Miquel circle, and it is the point of concurrence of eight remarkable lines of the complete quadrilateral, obviously including the four lines quoted above. 2. Some properties of the complete quadrilateral and notations Before we present our approach to the Hervey point, we recall some properties of the complete quadrilateral. Most of them were given by J. Steiner in 1827 [6]. An analysis of the Steiner’s note was recently published by J.-P. Ehrmann [1]. In what follows we quote the theorems as they were labelled by Steiner himself and reported in the Ehrmann’s review. They are referred to as theorems (S-n). The three theorems used in the course of the present paper are: Theorem S-1: The four lines of a complete quadrilateral form four associated triangles whose circumcircles pass through the same point (Miquel point M). Theorem S-2: The centers of the four circumcircles and the Miquel point M lie on the same circle (Miquel circle). Publication Date: February 4, 2011. Communicating Editor: Paul Yiu. 1Erroneously rendered “the Harvey point” in [5]. 2 A. Levelut Theorem S-4: The orthocenters of the four associated triangles lie on the same line (orthocenter or Miquel-Steiner line). In what follows the indices run from 1 to 4. The complete quadrilateral is formed with the lines dn. The associated trian- gle Tn is formed with three lines other than dn, and has circumcenter On and orthocenter is Hn. The four circumcenters are all distinct, otherwise the complete quadrilateral would be degenerate. According to Theorem S-2, the four centers On are concyclic on the Miquel circle with center O. The triangle Θn is formed with the three circumcenters other than On. It has circumcenter O and orthocenter is hn. The Euler line OnHn cuts the Miquel circle at On and another point Nn. The triangle MOnNn is inscribed in the Miquel circle. d4 d3 H3 N3 O1 O2 H1 M N4 d1 H4 O O4 N2 d2 N1 H2 O3 Figure 1. The Miquel circle and the Miquel-Steiner line It is worthwhile to mention two more properties: (1) The triangles Θn and Tn are directly similar [1]; (2) Each orthocenter hn is on the corresponding line dn. A note on the Hervey point of a complete quadrilateral 3 3. The Hervey point of a complete quadrilateral In order to introduce the Hervey point, we proceed in three steps, successively defining the orthocenter of a triangle, the pseudo-orthocenter of an inscriptable quadrangle and the Hervey point of a complete quadrilateral. We define the orthocenter H of triangle ABC by the means of the standard Sylvester relation [3, §416]: the vector joining the circumcenter O to the ortho- center H is equal to the sum of the three vectors joining O to the three vertices, namely, OH = OA + OB + OC. It is easy to see that the resultant AH of the two vectors OB and OC is perpen- dicular to the side BC. Therefore the line AH is the altitude drawn through the vertex A and, more generally, the point H is the point of concurrence of the three altitudes. The usual property of the orthocenter is recovered. We define the pseudo-orthocenter H of the inscriptable quadrangle with the fol- lowing extension of the Sylvester relation OH = OP1 + OP2 + OP3 + OP4, where P1, P2, P3, P4 are four distinct points lying on a circle with center O. We do not go further in the study of this point since many of its properties are similar to those of the Hervey point. We define the Hervey point h of a complete quadrilateral as the pseudo-orthocenter of the inscriptable quadrangle made up with the four concyclic circumcenters O1, O2, O3, O4 of the triangles T1, T2, T3, T4, which lie on the Miquel circle with center O. The extended Sylvester relation reads Oh = OO1 + OO2 + OO3 + OO4. The standard Sylvester relation used for the triangle Θn reads Ohn = X OOi. i6=n This leads to Oh = Ohn + OOn for each n = 1, 2, 3, 4. From these, we deduce Onh = Ohn, hnh = OOn and hnhm = OmOn for m =6 n. (1) The last relation implies that the four orthocenters hn are distinct since the centers On are distinct. Figure 2 shows the construction of the Hervey point starting with the orthocenter h4. 4 A. Levelut d4 d3 H3 h3 O1 M h H1 O2 h4 S h O4 2 H O d1 4 h1 H2 d2 O3 Figure 2. 4. Three theorems on the Hervey point Theorem 1. The Hervey point h is the center of the circle which bears the four orthocenters hn of the triangles Θn.