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Triangle Centres and Homogeneous Coordinates

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Triangle Centres and Homogeneous Coordinates A A E E F I Triangle Centres F H and Homogeneous B D C B D C Figure 1. Incentre of a triangle; Coordinates I is equidistant from the sides Figure 3. Orthocentre of an acute-angled triangle ClassRoom Incentre. Since the incentre is equidistant from them. So the reciprocals of the sides bear the same Part I - Trilinear Coordinates the sides, its trilinear coordinates are simply ratios to each other as the cosecant values. 1 : 1 : 1 (see Figure 1). Orthocentre. Now we turn our attention to the Centroid. Let G be the centroid of ABC (see orthocentre. We first consider the case ofan △ Figure 2). It is a well-known result of Euclidean acute-angled triangle ABC (see Figure 3). Here, geometry that triangles GAB, GBC and GCA are AD, BE, CF are perpendiculars from the vertices equal in area. If GD, GE and GF are A, B, C to the sides BC, CA, AB, respectively; H is perpendiculars to the sides, then GD a/2 = the orthocentre. · A Ramachandran GE b/2 = GF c/2 = k, say. Since HCD = 90◦ B, we get DHC = B, uring a course in Euclidean geometry at high school level, · · − GD = k/a GE = k/b GF = and sec B = HC/HD. Similarly, HCE = 90◦ a student encounters four classical triangle centres—the This yields: 2 , 2 , − / : : = / : / : / A, so EHC = A, and sec A = HC/HE. circumcentre, the incentre, the orthocentre and the 2k c, hence GD GE GF 1 a 1 b 1 c; Hence: Dcentroid (introduced as the points of concurrence of the these ratios form the trilinear coordinates of the centroid. HD HC HC sec A perpendicular bisectors of the sides, the bisectors of the angles of the = / = . triangle, the altitudes and the medians, respectively). We shall study Alternatively, the coordinates could be given as bc : HE HE HD sec C two alternative ways of describing and characterising these four ca : ab (multiplying through by abc), or as csc A : Similarly, HE/HF = sec B/sec C. Therefore the significant points. They are both known as homogeneous coordinate csc B : csc C. The last relation arises from the fact trilinear coordinates of the orthocentre are sec A : systems, but we explain the significance of this term later. In part Iof that the sides of a triangle bear the same ratios to sec B : sec C. the article, we consider the first of these: trilinear coordinates. each other as the sines of the angles opposite Let us see what happens as one of the angles (say A) approaches 90◦. The other two angles also Trilinear coordinates A approach limiting values which we assume are This approach was suggested by the German physicist-mathematician distinct from 0◦ and 90◦. Note that HE/HD = Julius Plücker in 1835 [1]. Here a triangle centre is characterised in cos A/cos B and HF/HD = cos A/cos C. As terms of its perpendicular distances from the three sides of the A 90◦, cos A 0; so HE 0 and HF 0 E → → → → triangle; or rather, the ratios of these distances. These ratios form the (in the limit, A, H, E, F coincide; see Figure 4). “trilinear coordinates” of the triangle centre. If ABC is the triangle F Two of the three quantities HD, HE, HF are now △ G and P the point in question, then the perpendicular distances zero, and it is customary to write the ratios as PD, PE, PF to the sides BC, CA, AB respectively are expressed as HD : HE : HF = 1 : 0 : 0. ratios involving the side lengths and/or trigonometric functions of the angles of the triangle. B D C It follows that for a right-angled triangle ABC with A = 90◦, the trilinear coordinates of the Figure 2. Centroid of an arbitrary triangle orthocentre are 1 : 0 : 0. Keywords: Triangle centre, incentre, centroid, orthocentre, circumcentre, trilinear coordinates, homogeneous coordinates, trigonometric ratio 4140 At Right Angles | Vol. 5, No. 1, March 2016 Vol. 5, No. 1, March 2016 | At Right Angles 4140 1 A A E E F I Triangle Centres F H and Homogeneous B D C B D C Figure 1. Incentre of a triangle; Coordinates I is equidistant from the sides Figure 3. Orthocentre of an acute-angled triangle ClassRoom Incentre. Since the incentre is equidistant from them. So the reciprocals of the sides bear the same Part I - Trilinear Coordinates the sides, its trilinear coordinates are simply ratios to each other as the cosecant values. 1 : 1 : 1 (see Figure 1). Orthocentre. Now we turn our attention to the Centroid. Let G be the centroid of ABC (see orthocentre. We first consider the case ofan △ Figure 2). It is a well-known result of Euclidean acute-angled triangle ABC (see Figure 3). Here, geometry that triangles GAB, GBC and GCA are AD, BE, CF are perpendiculars from the vertices equal in area. If GD, GE and GF are A, B, C to the sides BC, CA, AB, respectively; H is perpendiculars to the sides, then GD a/2 = the orthocentre. · A Ramachandran GE b/2 = GF c/2 = k, say. Since HCD = 90◦ B, we get DHC = B, uring a course in Euclidean geometry at high school level, · · − GD = k/a GE = k/b GF = and sec B = HC/HD. Similarly, HCE = 90◦ a student encounters four classical triangle centres—the This yields: 2 , 2 , − / : : = / : / : / A, so EHC = A, and sec A = HC/HE. circumcentre, the incentre, the orthocentre and the 2k c, hence GD GE GF 1 a 1 b 1 c; Hence: Dcentroid (introduced as the points of concurrence of the these ratios form the trilinear coordinates of the centroid. HD HC HC sec A perpendicular bisectors of the sides, the bisectors of the angles of the = / = . triangle, the altitudes and the medians, respectively). We shall study Alternatively, the coordinates could be given as bc : HE HE HD sec C two alternative ways of describing and characterising these four ca : ab (multiplying through by abc), or as csc A : Similarly, HE/HF = sec B/sec C. Therefore the significant points. They are both known as homogeneous coordinate csc B : csc C. The last relation arises from the fact trilinear coordinates of the orthocentre are sec A : systems, but we explain the significance of this term later. In part Iof that the sides of a triangle bear the same ratios to sec B : sec C. the article, we consider the first of these: trilinear coordinates. each other as the sines of the angles opposite Let us see what happens as one of the angles (say A) approaches 90◦. The other two angles also Trilinear coordinates A approach limiting values which we assume are This approach was suggested by the German physicist-mathematician distinct from 0◦ and 90◦. Note that HE/HD = Julius Plücker in 1835 [1]. Here a triangle centre is characterised in cos A/cos B and HF/HD = cos A/cos C. As terms of its perpendicular distances from the three sides of the A 90◦, cos A 0; so HE 0 and HF 0 E → → → → triangle; or rather, the ratios of these distances. These ratios form the (in the limit, A, H, E, F coincide; see Figure 4). “trilinear coordinates” of the triangle centre. If ABC is the triangle F Two of the three quantities HD, HE, HF are now △ G and P the point in question, then the perpendicular distances zero, and it is customary to write the ratios as PD, PE, PF to the sides BC, CA, AB respectively are expressed as HD : HE : HF = 1 : 0 : 0. ratios involving the side lengths and/or trigonometric functions of the angles of the triangle. B D C It follows that for a right-angled triangle ABC with A = 90◦, the trilinear coordinates of the Figure 2. Centroid of an arbitrary triangle orthocentre are 1 : 0 : 0. Keywords: Triangle centre, incentre, centroid, orthocentre, circumcentre, trilinear coordinates, homogeneous coordinates, trigonometric ratio 4140 At Right Angles | Vol. 5, No. 1, March 2016 Vol. 5, No. 1, March 2016 | At Right Angles 4140 1 A, H, E, F A A A F E F E E F B D C O B D C B D, O C O Figure 4. Orthocentre of a right-angled triangle Figure 7. Circumcentre of a right-angled triangle Figure 8. Circumcentre of an obtuse-angled triangle H B D C F obtained earlier; only, cos A has now assumed a Note from the editor: What is homogeneous zero value. about this system? The significance of the term In the case of an obtuse-angled triangle, the homogeneous may not be immediately apparent. In Figure 6. Circumcentre of an acute-angled triangle circumcentre O lies outside the triangle (see ‘ordinary’ coordinate geometry, the equation of a E + + = , , Figure 8, where A is obtuse). line has the form ax by c 0, where a b c A are constants. Note that this equation is not and we find that the inherent symmetry ofthe Here we have BOD = 180◦ A, so OD = − homogeneous: two terms have degree 1, while one formula has been restored. R cos A. The relations OE = R cos B and OF = − term has degree 0. Similarly, the question of a B D C R cos C remain unchanged. So we get OD : OE : circle has the form x2 + y2 + 2gx + 2fy + c = 0; Circumcentre. Next, we consider the OF = cos A : cos B : cos C. If we adopt the this too is not homogeneous. In some settings, it Figure 5. Orthocentre of an obtuse-angled triangle − circumcentre.
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