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In Search of More Triangle Centres. a Source of Classroom Projects in Euclidean Geometry

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In Search of More Triangle Centres. a Source of Classroom Projects in Euclidean Geometry International Journal of Mathematical Education in Science and Technology, Vol. 36, No. 8, 2005, 889–912 In search of more triangle centres. A source of classroom projects in Euclidean geometry S. ABU-SAYMEH and M. HAJJA* Department of Mathematics, Yarmouk University, Irbid – Jordan (Received 25 June 2004) A point E inside a triangle ABC can be coordinatized by the areas of the triangles EBC, ECA, and EAB. These are called the barycentric coordinates of E. It can also be coordinatized using the six segments into which the cevians through E divide the sides of ABC, or the six angles into which the cevians through E divide the angles of ABC, or the six triangles into which the cevians through E divide ABC, etc. This article introduces several coordinate systems of these types, and investigates those centres of ABC whose coordinates, relative to a given coordinate system, are linear (or quasi-linear) with respect to appropriate elements of ABC, such as its side-lengths, its angles, etc. This results in grouping known centres into new families, and in discovering new centres. It also leads to unifying several results that are scattered in the literature, and creates several open questions that may be suitable for classroom discussions and team projects in which algebra and geometry packages are expected to be useful. These questions may also be used for Mathematical Olympiad training and may serve as supplementary material for students taking a course in Euclidean geometry. 1. Introduction and terminology Throughout this paper, the letters A, B,andC stand for the measures of the angles of a triangle ABC, and the letters a, b, and c stand for its side-lengths BC, CA, and AB, respectively. The area of a triangle XYZ is denoted by ½XYZ, and its perimeter by p(XYZ). We draw the reader’s attention to the fact that some letters stand for different things in different sections. This holds in particular for the letters x, y, z, X, Y,andZ, which are used in almost all the sections. The sections in this paper are self-contained, and thus they can be read in any order. For terms that are used without being defined, and generally for any terms pertaining to triangle geometry, the reader is referred to the encyclopedic work [1], or to the standard textbook [2]. Let ABC be a general triangle, and let Á denote its area ½ABC. For a point E inside ABC, the barycentric or areal coordinates x, y,andz of E are defined by ½EBC ½ECA ½EAB x ¼ , y ¼ , z ¼ Á Á Á *Corresponding author. Email: [email protected] International Journal of Mathematical Education in Science and Technology ISSN 0020–739X print/ISSN 1464–5211 online # 2005 Taylor & Francis http://www.tandf.co.uk/journals DOI: 10.1080/00207390500137928 890 S. Abu-Saymeh and M. Hajja and the trilinear coordinates , and of E are defined as the altitudes from E of the triangles EBC, ECA and EAB. Thus the barycentric and trilinear coordinates of E are related by 2xÁ ¼ a,2yÁ ¼ b,2zÁ ¼ c ð1Þ The barycentric and trilinear coordinates can be defined for all points in the plane of ABC by adopting the convention that ½EBC is taken positive or negative according as E and A lie on the same side or on different sides of the line BC, and so on. A similar convention is adopted for trilinear coordinates, and equations (1) are retained. Note that a point is interior if and only if its barycentric (equivalently, trilinear) coordinates are positive. Note also that a triple of real numbers ðx, y, zÞ qualifies as the barycentric coordinates of some point if and only if x þ y þ z ¼ 1. Both barycentric and trilinear coordinates are powerful tools in studying the geometry of the triangle. It is also transparent how to define them for simplices in any dimension. The centroid of ABC is the point where the medians meet. It is also the centre of mass of three equal point masses placed at the vertices, and is also the centre of mass of the triangular lamina ABC with uniform density. The centre of mass of the triangular frame ABC having uniform density is a different point usually referred to as the Spieker centre. The incentre of ABC is the centre of its incircle. The Gergonne centre is the point of concurrence of the cevians AA0, BB0, and CC0, where A0, B0, and C0 are the points where the incircle touches the sides. Similarly, the Nagel centre is the point of concurrence of the cevians AA00, BB00,andCC00, where A00, B00, and C00 are the points where the excircles touch the sides. That such lines do indeed concur follows from Ceva’s Theorem according to which the cevians AA0, BB0, and CC0 concur if and only if AC0 BA0 CB0 ¼ 1 C0B A0C B0A Other centres will be mentioned in the sequel. For the time being, these centres were selected to illustrate certain properties of their barycentric coordinates. It is obvious that the barycentric coordinates of the centroid are ð1=3, 1=3, 1=3Þ, and it is not difficult to see that the barycentric coordinates of the incentre, the Spieker centre, and the Nagel centre are given by a b þ c Àa þ b þ c , À , À , , À , À , , À , À a þ b þ c 2ða þ b þ cÞ a þ b þ c where the second and third coordinates are filled in by applying the permutation ða ° b ° c ° aÞ, i.e. the cycle ðabcÞ. The reader is invited to find the barycentric coordinates of the Gergonne centre and to see that they cannot be expressed in the form f / g, where f and g are linear forms in a, b, and c. The family of centres whose barycentric coordinates can be so expressed is studied in [3, Theorem 2 and Note 3], where it is shown to form a straight line that turns out to be the first central line in the list of [1, p. 128]. In the next sections, we introduce other types of coordinates, and we investigate centres whose such coordinates can be nicely expressed. In search of more triangle centres 891 2. Side-intercept coordinates: the centroid, the Gergonne centre, the Nagel centre and more Let ABC be a general triangle, and let A0, B0, and C0 be points on its sides. Let x, X, y, Y, z, and Z be defined (as shown in figure 1) by x ¼ BA0, y ¼ CB0, z ¼ AC0 ð2Þ X ¼ a À x, Y ¼ b À y, Z ¼ c À z ð3Þ Ceva’s theorem states that the cevians AA0, BB0, CC0 are concurrent if and only if xyz ¼ XYZ ð4Þ The point E of concurrence is completely determined by the triple ðx, y, zÞ, which in turn is arbitrary except for having to satisfy the Ceva condition xyz ¼ða À xÞðb À yÞðc À zÞ. Following [3], we call ðx, y, zÞ the side-intercept coordi- nates of E. Like the barycentric and trilinear coordinates, the side-intercept coordi- nates are allowed to be negative by thinking of BA0, CB0, and AC0 in (2) to be directed segments whose lengths change sign when their directions change. Thus for example, if A0 is such that B lies between C and A0, then BA0 is negative. With this understanding, everything above remains valid, and a point is interior if its side-intercept coordinates are positive. The most natural conditions that satisfy (4) are given by the following three systems of requirements: x ¼ X, y ¼ Y, z ¼ Z ð5Þ z ¼ Y, x ¼ Z, y ¼ X ð6Þ y ¼ Z, z ¼ X, x ¼ Y ð7Þ A z Y B C E Z y B xXA C Figure 1. 892 S. Abu-Saymeh and M. Hajja (Possibilities such as ðx, y, zÞ¼ðX, Z, YÞ are excluded since they are satisfied only if ABC is isosceles.) It is clear that (5) and (6) define the centroid and the Gergonne centre of ABC, respectively, and it is not difficult to see that (7) defines the Nagel centre. We remind the reader that the Gergonne (respectively, the Nagel) centre is customarily defined by the equivalent geometric requirement that A0, B0, and C0 are the points where the sides touch the incircle (respectively, the excircles) [1]. Using (3), one finds that the solutions of (5), (6), and (7) are given by a b c x ¼ , y ¼ , z ¼ 2 2 2 a À b þ c b À c þ a c À a þ b x ¼ , y ¼ , z ¼ 2 2 2 a þ b À c b þ c À a c þ a À b x ¼ , y ¼ , z ¼ 2 2 2 in the same order. The three centres mentioned above are symmetric in the sense that their side- intercept coordinates are invariant under any relabelling of the vertices A, B, and C. To state this algebraically, we note that every permutation on fA, B, Cg (together with the natural permutation it induces on fA0, B0, C0g) gives rise to a permutation on each of the sets fx, y, zg, fX, Y, Zg,andfa, b, cg in accordance with (2) and (3) above. Thus the cyclic permutations ðBCÞ and ðABCÞ stand for the permutations ðbcÞðxXÞðyZÞðzYÞ and ðabcÞðxyzÞðXYZÞ, respectively. With this understanding, the symmetry of a centre amounts to the requirement that the system of equations that define its coordinates is invariant under every permutation of fA, B, Cg. Here, a system (Ã) is invariant under a permutation if the system obtained by applying to (Ã) has the same solution as (Ã).
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