Elem. Math. 57 00) 6 1 001-6018/0/01006-6 $ 1.50+0.0/0 c irkhäuser Verlag, asel, 00 Elemente der Mathematik The notable configuration of inscribed equilateral triangles in a triangle las Herrera Gómez las Herrera Gómez obtained his Ph.D. in mathematics at the University utònoma of arcelona in 1994. Presently, he is a professor at the University Rovira i Virgili of Tarragona. His main fields of interest are: the geometry of foliations, the dynamics of galaxies, classical geometry, and the mechanics of fluids and turbulence. 1 Introduction Many notable configurations for a given triangle have been described by different geometers in the past. mong the most famous classic configurations are the Euler and Simson lines, the Gergonne, Lemoine, rocard and Nagel concurrences, the Feuerbach, rocard, Lemoine, Tuker and Taylor circumferences, etc. More modern configurations can be found in the Soddy circles [9], the Euler-Gergonne-Soddy triangle [8], the Torricelli configuration [10], etc. In fact, one can see in [] four hundred configurations triangle centers), in [4] even eight hundred configurations are picked up. Recently, new configurations continue to appear, for example Morley triangles related to the Feuerbach circumferences [5], associated rectangular hyperbolas [1], triangle centers associated with a rhombus [6], Euler lines concurrent on the nine-point circle [7], etc. mong modern configurations we focus our attention on the Lucas circles. Edouard Lucas studied the configuration given by three inscribed squares in a triangle with a side parallel to each one of the sides of the triangle. With them appear three circles. Recently [], Yiu and Hatzipolakis have been continuing the study of this configuration and, in particular, have shown recently that the Lucas circles are tangent to each other. ekanntlich gibt es eine Vielzahl bereits untersuchter Konfigurationen zu einem gegebenen ebenen Dreieck. Der utor des nachfolgenden eitrags knüpft an eine von E. Lucas studierte Konfiguration an, die ein Dreieck mit drei einbeschriebenen Quadraten zum usgangspunkt hat und zu den sogenannten Lucas-Kreisen führt. Hier wird nun ein Dreieck mit drei einbeschriebenen gleichseitigen Dreiecken, von denen jeweils genau eine Seite zu einer der Seiten des usgangsdreiecks parallel ist, untersucht. ei dieser Konfiguration zeigt sich unter anderem, dass jeweils vier der insgesamt neun Eckpunkte der einbeschriebenen Dreiecke auf drei Kreisen liegen, die sich in einem Punkt schneiden..
Elem. Math. 57 00) 7 In this paper we take on Lucas idea and consider not the three inscribed squares, but the three inscribed equilateral triangles with one side parallel to each one of the sides of thetriangleseefig.1). The notable configuration which appears is outlined below and will be summarized in the theorem of the following section. We will show that the centers of these equilateral triangles are aligned see Fig. ). We will see also that the mid-points of the sides of the equilateral triangles are aligned three by three see Fig. ). We will find that the simple ratios of these mid-points and the simple ratios of the vertices are equal to the simple ratio of the centers. ut the most remarkable feature will be the appearance of three circumferences which link the vertices four by four, and these three circles are concurrent in a point see Figs., 4). 1 1 1 1 1 1 Fig. 1 onstruction of the three inscribed equilateral triangles. onfiguration First of all, in order to establish our notations we recall that the simple ratio,, ) of three aligned points is,, ) =± with a positive sign if is not between and, and a negative sign otherwise. We recall that in order to construct an inscribed equilateral triangle in with a side parallel to side, we consider the exterior equilateral triangle with side,. Let be the intersection of the line with the line. The lines through the point parallel to and intersect with the lines and at the points 1 and respectively. The triangle 1 is the inscribed equilateral triangle see Fig. 1). The other two triangles 1 and 1, with sides parallel to and respectively, can be constructed in a similar way. The description of the configuration can be summarized in the following theorem.
8 Elem. Math. 57 00) Fig. The centers O, O, O are aligned. The lines,, are concurrent outward Fermat point F = X 1 ). The mid-points 1, 1, 1 are aligned; the mid-points 1, 1, are aligned; the midpoints, 1, are aligned. ll the simple ratios coincide with the simple ratio O, O, O ). Theorem 1 Let be a triangle. Let 1, 1, 1 be the three inscribed equilateral triangles with 1 parallel to, 1 parallel to, 1 parallel to and 1, 1, on the line. Let O, O, O be the respective centers of the three equilateral triangles, and let ij, ij, ij be the respective mid-points of the sides i j, i j, i j. Then we have: 1. The center points O, O, O are aligned.. The lines,, are concurrent in the outward Fermat point).. The mid-points 1, 1, 1 are aligned, the mid-points 1, 1, are aligned and the mid-points, 1, are also aligned. 4. The following simple ratios are equal: O, O, O )=,, 1 )= 1, 1, )=,, ) = 1, 1, 1 )= 1, 1, )=,, 1 ). 5. The points,, 1, are concyclic, the points,,, are concyclic and the points, 1, 1, are also concyclic. 6. The above circumferences 1,, 1 1 are concurrent in a point η. Proof. Take a system of artesian coordinates in such a way that =a, b) with 0 a, b 1, =0, 0), =1, 0). straightforward computation proves the parts 1 and of the theorem see Fig. ). Part of the theorem is well-known. The concurrence of the lines,, is the outward Fermat point F X 1 in [4]).
Elem. Math. 57 00) 9 1 1 1 η Fig. The points,, 1 and are concyclic, the points,, and are concyclic and the points, 1, 1 and are concyclic. lso, the three circumferences are concurrent at the point η. Part 4 can be found with a straightforward computation. The value of the simple ratio obtaining in every case is the same expression: O, O, O )= 1 a ) b a 6a + b + ) b + b + ) a a + b ). b Part 5 see Figs., 4) can be found with a long computation and we have that the points,, 1, belong to the circumference of center O given by O x = 1 a + a b + ab + a + ab + b + 5b a + b + ) b b + ), O y = 1 a 5 a b + ab a + ab 5 b 9b a + b + ) b b + ), and radius r : a +b )a 4 4 a b+10a b 4 ab 6a +8 a b 14ab +a 4 ab+7b 4 +8 b +7b ) a +b + b) b+. ) nalogously, the points,,, belong to the circumference of center O given by: O x = 1 a + a b + ab 6a + a + b + 6b + b a 6a + b + ) b + b + ), O y = 1 a + 5 ab + ab 6a 8 ab + a + 5 b + 6b + b a 6a + b + ) b + b + ),
0 Elem. Math. 57 00) 1 η 1 1 Fig. 4 oncurrence of the three circumferences. and radius r : a a+b +1)a 4 +4 a b+10a b +4 ab 6a 4 a b 6ab +a +7b 4 +4 b +b ) a 6a+b + b+) b+. ) nalogously the points 1,,, 1 belong to the circumference of center O given by: a 4 +4 a b+6a b +4 ab 6a a b ab +a + ab+b 4 + b +5b O x = a + b+b )a 6a++ b+b ) O y = 1 a a + b + ) b a ) b ) b 1 + a a + b + ) b a 6a + b + ), b +, and radius r : 9 a +b )a a+b +1)a 4 +6a b 6a 4 a b 6ab +a +4 ab+b 4 +4 b +7b ) a +b + b) a 6a+b +. b+) Finally, part 6 corresponds to the above circumferences 1,, 1 1 are concurrent in a point η see Figs., 4). To prove this statement let η be the image of point under the axial symmetry with respect to the line O O. We obtain ) 1 ς η = τ, 1 υ 6 φ
Elem. Math. 57 00) 1 with ς = a 6 + 8 a 5 b + 9a 4 b + 16 a b + 9a b 4 + 8 ab 5 a 5 11 a 4 b 6a b 14 a b ab 4 a 4 a b 10 ab + a + 5 a b + ab + b 6 b 5 5b 4 + 5 b, τ = a 6 + 9a 4 b + 9a b 4 + 9 a 4 b 9a 5 18a b + 10 a b 9ab 4 + 1a 4 18 a b + 18a b 10 ab 9a + 9 a b + a 9ab + b 6 + b 5 b 4 + b + b, υ = a a + b a b) 4 a b 6a + ab + a 5b 4 + b ), φ = a 6 + 9a 4 b + 9a b 4 9a 5 + 9 a 4 b 18a b + 10 a b 9ab 4 + 1a 4 18 a b + 18a b 10 ab 9a 9ab + 9 a b + a + b 6 + b 5 b 4 + b + b. This point η, by construction, belongs to the two circumferences 1,. It only remains to prove that η belongs to the circumference 1 1. ut when we compute the distance between η and O we obtain exactly the same expression obtained above for r. This completes the proof. cknowledgement I would like to thank gustí Reventós Tarrida, Departament de Matemàtiques, Universitat utònoma de arcelona for his constructive criticism of this paper. References [1] erin, Z.: On Properties of Rectangular Hyperbolas. Geom. Dedicata 84 001), 41 47. [] Hatzipolakis,.P., Yiu, P.: The Lucas ircles of a Triangle. mer. Math. Monthly 108 001), 444 446. [] Kimberling,.: Triangle enters and entral Triangles. ongressus Nunerantium 19 1998). [4] Kimberling,.: Encyclopedia of Triangle enters. http://cedar.evansville.edu/ ck6/encyclopedia/. [5] van Lamoen, F.: Morley Related Triangles on the Nine-Point ircle. mer. Math. Monthly 107 000), 941 945. [6] van Lamoen, F.: Triangle enters ssociated with Rhombi. Elem. Math. 55 000), 10 109. [7] van Lamoen, F.: Euler Lines oncurrent on the Nine-Point ircle. Problem 10796 [000,67]. mer. Math. Monthly 108 001), 569. [8] Oldknow,.: The Euler-Gergonne-Soddy Triangle of a Triangle. mer. Math. Monthly 10 1996), 19 9. [9] Vandeghen,.: Soddy s ircles and the de Longchamps Point of a Triangle. mer. Math. Monthly 71 1964), 176 179. [10] Wetzel, J.E.: onverses of Napoleon s Theorem. mer. Math. Monthly 99 199), 9 51. las Herrera Gómez Departament d Enginyeria Informàtica i Matemàtiques Universitat Rovira i Virgili 4007 Tarragona, Spain e-mail: bherrera@etse.urv.es