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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. Bradley and Geoff C. Smith, The locations of triangle centers,57 Christopher J. Bradley and Geoff C. Smith, The locations of the Brocard points,71 Floor van Lamoen, Archimedean adventures,79 Anne Fontaine and Susan Hurley, Proof by picture: Products and reciprocals of diagonals length ratios in the regular polygon,97 Hiroshi Okumura and Masayuki Watanabe, A generalization of Power’s Archimedean circles, 103 Stanley Rabinowitz, Pseudo-incircles, 107 Wilson Stothers, Grassmann cubics and desmic structures, 117 Matthew Hudelson, Concurrent medians of (2n +1)-gons, 139 Dieter Ruoff, On the generating motions and the convexity of a well-known curve in hyperbolic geometry, 149 Mowaffaq Hajja, A very short and simple proof of “the most elementary theorem” of euclidean geometry, 167 S´andor Kiss, The orthic-of-intouch and intouch-of-orthic triangles, 171 Kurt Hofstetter, A 4-step construction of the golden ratio, 179 Eisso J. Atzema, A theorem by Giusto Bellavitis on a class of quadrilaterals, 181 Wladimir Boskoff and Bogdan Suceav˘a, A projectivity characterized by the Pythagorean relation, 187 Bogdan Suceav˘a and Paul Yiu, The Feuerbach point and Euler lines, 191 Eric Danneels, A simple perspectivity, 199 Quang Tuan Bui, Pedals on circumradii and the Jerabek center, 205 Bernard Gibert, The Simmons conics, 213 Nikolaos Dergiades, A synthetic proof and generalization of Bellavitis’ theorem, 225 Huub van Kempen, On some theorems of Poncelet and Carnot, 229 Charles Thas, The Droz-Farny theorem and related topics, 235 Zvonko Cerin,ˇ On butterflies inscribed in a quadrilateral, 241 Eric Danneels, On triangles with vertices on the angle bisectors, 247 Matthew Hudelson, Formulas among diagonals in the regular polygon and the Catalan numbers, 255 Khoa Lu Nguyen, A note on the barycentric square root of Kiepert perspectors, 263 Clark Kimberling, Translated triangle perspective to a reference triangle, 269 James L. Parish, On the derivative of a vertex polynomial, 285 Alexei Myakishev, On two remarkable lines related to a quadrilateral, 289 Max A. Tran Intersecting circles and their inner tangent circle, 297 K. R. S. Sastry, Two Brahmagupta problems, 301 Floor van Lamoen, Square wreaths around hexagons, 311 Jean-Pierre Ehrmann, Some geometric constructions, 327 Amy Bell, Hansen’s right triangle theorem, its converse and a generalization, 335 Paul Yiu, Some constructions related to the Kiepert hyperbola, 343 Author Index, 359 Forum Geometricorum b Volume 6 (2006) 1–16. bbb FORUM GEOM ISSN 1534-1178 On Mixtilinear Incircles and Excircles Khoa Lu Nguyen and Juan Carlos Salazar Abstract. A mixtilinear incircle (respectively excircle) of a triangle is tangent to two sides and to the circumcircle internally (respectively externally). We study the configuration of the three mixtilinear incircles (respectively excircles). In particular, we give easy constructions of the circle (apart from the circumcircle) tangent the three mixtilinear incircles (respectively excircles). We also obtain a number of interesting triangle centers on the line joining the circumcenter and the incenter of a triangle. 1. Preliminaries In this paper we study two triads of circles associated with a triangle, the mix- tilinear incircles and the mixtilinear excircles. For an introduction to these circles, see [4] and §§2, 3 below. In this section we collect some important basic results used in this paper. Proposition 1 (d’Alembert’s Theorem [1]). Let O1(r1), O2(r2), O3(r3) be three circles with distinct centers. According as ε =+1or −1, denote by A1ε, A2ε, A3ε respectively the insimilicenters or exsimilicenters of the pairs of circles ((O2), (O3)), ((O3), (O1)), and ((O1), (O2)).Forεi = ±1, i =1, 2, 3, the points A1ε1 , A2ε2 and A3ε3 are collinear if and only if ε1ε2ε3 = −1. See Figure 1. The insimilicenter and exsimilicenter of two circles are respectively their inter- nal and external centers of similitude. In terms of one-dimensional barycentric coordinates, these are the points r2 · O1 + r1 · O2 ins(O1(r1),O2(r2)) = , (1) r1 + r2 −r2 · O1 + r1 · O2 exs(O1(r1),O2(r2)) = . (2) r1 − r2 Proposition 2. Let O1(r1), O2(r2), O3(r3) be three circles with noncollinears cen- ters. For ε = ±1, let Oε(rε) be the Apollonian circle tangent to the three circles, all externally or internally according as ε =+1or −1. Then the Monge line con- taining the three exsimilicenters exs(O2(r2),O3(r3)), exs(O3(r3),O1(r1)), and exs(O1(r1),O2(r2)) is the radical axis of the Apollonian circles (O+) and (O−). See Figure 1. Publication Date: January 18, 2006. Communicating Editor: Paul Yiu. The authors thank Professor Yiu for his contribution to the last section of this paper. 2 K. L. Nguyen and J. C. Salazar A3− A2− O1 A2+ A3+ O+ O− A1− O2 A1+ O3 Figure 1. Lemma 3. Let BC be a chord of a circle O(r). Let O1(r1) be a circle that touches BC at E and intouches the circle (O) at D. The line DE passes through the midpoint A of the arc BC that does not contain the point D. Furthermore, AD · AE = AB2 = AC2. Proposition 4. The perspectrix of the circumcevian triangle of P is the polar of P with respect to the circumcircle. Let ABC be a triangle with circumcenter O and incenter I. For the circumcircle and the incircle, r · O + R · I ins((O), (I)) = = X55, R + r −r · O + R · I exs((O), (I)) = = X56. R − r in the notations of [3]. We also adopt the following notations. A0 point of tangency of incircle with BC A1 intersection of AI with the circumcircle A2 antipode of A1 on the circumcircle Similarly define B0, B1, B2, C0, C1 and C2. Note that (i) A0B0C0 is the intouch triangle of ABC, (ii) A1B1C1 is the circumcevian triangle of the incenter I, On mixtilinear incircles and excircles 3 (iii) A2B2C2 is the medial triangle of the excentral triangle, i.e., A2 is the midpoint between the excenters Ib and Ic. It is also the midpoint of the arc BAC of the circumcircle. 2. Mixtilinear incircles The A-mixtilinear incircle is the circle (Oa) that touches the rays AB and AC at Ca and Ba and the circumcircle (O) internally at X. See Figure 2. Define the B- and C-mixtilinear incircles (Ob) and (Oc) analogously, with points of tangency Y and Z with the circumcircle. See [4]. We begin with an alternative proof of the main result of [4]. Proposition 5. The lines AX, BY , CZ are concurrent at exs((O), (I)). Proof. Since A =exs((Oa), (I)) and X =exs((O), (Oa)), the line AX passes through exs((O), (I)) by d’Alembert’s Theorem. For the same reason, BY and CZ also pass through the same point. A2 Y A A B1 B0 O b C1 O O c Ba Ba Z I C0 I Ca Oa Ca Oa B C B A0 C X X A1 Figure 2 Figure 3 Lemma 6. (1) I is the midpoint of BaCa. = r (2) The A-mixtilinear incircle has radius ra cos2 A . 2 (3) XI bisects angle BXC. See Figure 3. Consider the radical axis a of the mixtilinear incircles (Ob) and (Oc). Proposition 7. The radical axis a contains (1) the midpoint A1 of the arc BC of (O) not containing the vertex A, (2) the midpoint Ma of IA0, where A0 is the point of tangency of the incircle with the side BC. 4 K. L. Nguyen and J. C. Salazar Y A B1 Bc Cb B0 O C1 b Mb Oc Z C0 I Mc Ma B Ac A0 Ab C A1 Figure 4. Proof. (1) By Lemma 3, Z, Ac and A1 are collinear, so are Y , Ab, A1. Also, 2 2 A1Ac · A1Z = A1B = A1C = A1Ab · A1Y . This shows that A1 is on the radical axis of (Ob) and (Oc). (2) Consider the incircle (I) and the B-mixtilinear incircle (Ob) with common ex-tangents BA and BC. Since the circle (I) touches BA and BC at C0 and A0, and the circle (Ob) touches the same two lines at Cb and Ab, the radical axis of these two circles is the line joining the midpoints of CbC0 and AbA0.
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