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CIRCUMCEVIAN INVERSION PERSPECTOR a Generalisation of the Triangle Center X(35)

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CIRCUMCEVIAN INVERSION PERSPECTOR a Generalisation of the Triangle Center X(35) 11/22/2019 CIRCUMCEVIAN INVERSION PERSPECTOR A Generalisation of the Triangle Center X(35) SUREN [email protected] CIRCUMCEVIAN INVERSION PERSPECTOR A Generalization of the Triangle Center X(35) SUREN [email protected] ABSTRACT. The triangle center 푋(35) as mentioned in the Kimberling’s Encyclopedia of Triangle Centers, is the point of concurrence of the lines joining the vertices of the triangle to the inverses of the opposite excentres in the circumcircle. In this paper, we obtain a generalization of the same. NOTATION Let 퐴퐵퐶 be a triangle. We denote its side lengths 퐵퐶, 퐶퐴 and 퐴퐵 by 푎, 푏, 푐 , its perimeter by 2푠, 푠 being the semiperimeter, and its area by 훥. Its well-known triangle centers are the circumcenter 푂, the incenter 퐼, the orthocenter 퐻 and the centroid 퐺. The nine-point center (푁) is the center of the nine-point circle and is the midpoint of segment 푂퐻. 퐼푎 represents the excenter opposite to 퐴 and define 퐼푏 and 퐼푐 cyclically. The inradius is represented by 푟 and the circumradius by 푅. The Euler Line is the line on which the circumcenter, the centroid and the orthocenter lie. 푁푎 represents the Nagel Point, i.e. the point of concurrence of the lines joining the vertices of the triangle to the points of tangency of the corresponding excircles to the opposite sides. (퐴퐵퐶) represents the circumcircle of triangle 퐴퐵퐶. 푃표푤휔(푋) represents the power of the point 푋 wrt the circle ω. The ordered triple 푥: 푦: 푧 represents the point with barycentric coordinates 푥 푦 푧 ( , , ) 푥 + 푦 + 푧 푥 + 푦 + 푧 푥 + 푦 + 푧 푓(푎, 푏, 푐): 푓(푏, 푐, 푎): 푓(푐, 푎, 푏) can be represented as 푓(푎, 푏, 푐) ∷. The point 푋(푘) represents the triangle center 푋(푘) as listed in the Kimberling’s Encyclopedia of Triangle Centers. CONWAY’S NOTATION 2 2 2 (푏 + 푐 − 푎 )⁄ 푆 = 2훥 and 푆퐴 = 2 = 푏푐 cos 퐴. 푆퐵 and 푆퐶 are defined cyclically. 2 ( 2 2) a + b + c ⁄ 푆휔 = 푆 cot 휔 , cot(ω) = 2푆 , where 휔 is the Brocard angle. Throughout this paper, the above notation shall be used. The symbols have the same meaning as in this section unless specified otherwise. THE TRIANGLE CENTER 푋(35) Page | 1 The triangle center 푋(35) as mentioned in the Kimberling’s Encyclopedia of Triangle Centers, is the point of concurrence of the lines joining the vertices of the triangle to the inverses of the opposite excentres in the circumcircle. P’ Figure 1. Triangle Center X(35) THE INCENTER-EXCENTER LEMMA We begin with the well-known Incenter-Excenter Lemma that relates the incenter and excenters of a triangle. Lemma. Let 퐴퐵퐶 be a triangle. Denote by 퐿 the mid-point of arc 퐵퐶 not containing 퐴. Then, 퐿 is the center of the circle passing through 퐼, 퐼푎, 퐵, 퐶. Proof. Note that the points 퐴, 퐼, 퐼푎 and 퐿 are collinear (as 퐿 lies on the angle bisector of ∠퐵퐴퐶). We wish to show that 퐿퐵 = 퐿퐼 = 퐿퐶 = 퐿퐼푎. 1 1 First, notice that ∠퐿퐵퐼 = ∠퐿퐵퐶 + ∠퐶퐵퐼 = ∠퐿퐴퐶 + ∠퐶퐵퐼 = ∠퐼퐴퐶 + ∠퐶퐵퐼 = ⁄2 ∠퐴 + ⁄2 ∠퐵 . 1 1 However, ∠퐵퐼퐿 = ∠퐵퐴퐼 + ∠퐴퐵퐼 = ⁄2 ∠퐴 + ⁄2 ∠퐵 = ∠퐿퐵퐼. Thus, triangle 퐵퐼퐿 is isosceles with 퐿퐵 = 퐿퐼. Similarly, 퐿퐼 = 퐿퐶. Hence, 퐿 is the circumcenter of triangle 퐼퐵퐶. The circle passing through 퐼, 퐵 and 퐶 must pass through 퐼푎 as well (the quadrilateral 퐼퐶퐼푎퐵 is cyclic as the opposite angles add up to 180°). Thus, 퐿 is the center of the circle passing through 퐼, 퐼푎, 퐵, 퐶. Using this property that relates the excentres and the incenter, we try to obtain a generalization of the triangle center 푋(35). Page | 2 Figure 2. The Incenter-Excentre Lemma THE CIRCUMCEVIAN-INVERSION PERSPECTOR The following theorem is the crux of this paper. Theorem (Circumcevian-inversion perspector). The triangle formed by the inverses of reflections of a point in the vertices of its circumcevian triangle is perspective to the original triangle at a point called the Circumcevian- inversion perspector of that point with respect to the original triangle. P’ Figure 3. The Circumcevian-inversion perspector 푃′ Page | 3 In simple words, let 푃 be any point in the plane of a triangle 퐴퐵퐶 and the lines 퐴푃, 퐵푃 and 퐶푃 intersect (퐴퐵퐶) again in 퐷, 퐸 and 퐹 respectively. The reflection of 푃 in 퐷, 퐸 and 퐹 are 푋, 푌 and 푍. The inverses of 푋, 푌 and 푍 in (퐴퐵퐶) are the points 푋′, 푌′ and 푍′ respectively. Then the lines 퐴푋′, 퐵푌′ and 퐶푍′ are concurrent at a point which we shall call throughout this paper, the ‘Circumcevian-inversion perspector’ of 푃 . The properties of the Circumcevian-inversion perspector of a point are discussed in the later sections. Proof. Construct the circles (퐴푂푋), (퐵푂푌), (퐶푂푍). 푃표푤(퐴푂푋)(푃) = 퐴푃 ∙ 푃푋 = 퐴푃 ∙ 2푃퐷 = 2푃표푤(퐴퐵퐶)(푃) Similarly, we can prove that 푃표푤(퐵푂푌)(푃) = 푃표푤(퐶푂푍)(푃) = 2푃표푤(퐴퐵퐶)(푃). So, the power of 푃 and 푂 wrt the three circles is equal. This implies that the three circles, (퐴푂푋), (퐵푂푌), (퐶푂푍) are coaxial and their radical axis is the line 푂푃. So, these circles share a second common point (other than 푂), say 퐾. Define 퐼 as the inversion about the circle (퐴퐵퐶). The image of any point 푋 under this inversion is 퐼(푋). Under the inversion, (퐴푂푋) is mapped to the line 퐴푋′, (퐵푂푌) is mapped to the line 퐵푌′, (퐶푂푍) is mapped to the line 퐶푍′. The lines 퐴푋′, 퐵푌′, and 퐶푍′ share the points 퐼(푂) and 퐼(퐾). Hence the lines 퐴푋′, 퐵푌′, and 퐶푍′ are concurrent at the point 퐼(퐾). (The other common point i.e. 퐼(푂) is a point at infinity). Remark. 푋, 푌 and 푍 can also be understood as the images of the point 푃 in the homothety with factor −1 centered at the points 퐷, 퐸 and 퐹. If the factor of homothety is changed to some other number, in that case as well, the lines 퐴푋′, 퐵푌′ and 퐶푍′ are concurrent. The above proof with minor modifications works. If the factor of homothety is changed to ℎ then, this perspector is called the ℎ-circumcevian-inversion perspector of 푃 with respect to the original triangle. The (−1) -circumcevian-inversion perspector is simply referred to as the circumcevian-inversion perspector. It is clear by the incenter-excenter lemma that the triangle center 푋(35) is the Circumcevian-inversion perspector of the incenter. Also, the circumcenter is the Circumcevian-inversion perspector of itself. PROPERTIES OF THE CIRCUMCEVIAN-INVERSION PERSPECTOR OF A POINT The Circumcevian-inversion perspector of a point displays interesting properties, three of which are listed below. The Circumcevian-inversion perspector opens the windows for many more triangle centers to be discovered. Property 1. The Circumcevian-inversion perspector of the point 푃 wrt triangle 퐴퐵퐶 lies on the line 푃푂, 푂 being the circumcenter of 퐴퐵퐶. Proof. Since the point 퐾 lies on the line 푃푂, 퐼(퐾) must lie on 푃푂 as well. ′ 푃푃′ Property 2. If the distance 푃푂 = 푑, and 푃 is the Circumcevian-inversion perspector of 푃, then ⁄푃′푂 = 1 − 2 푑 ⁄ 푅2. Proof. 2 2 푃표푤(퐴퐵퐶)(푃) = 푅 − 푑 = 퐴푃 ∙ 푃퐷 2 2 푃표푤(퐴푂푋)(푃) = 퐴푃 ∙ 푃푋 = 2퐴푃 ∙ 푃퐷 = 2( 푅 − 푑 ) = 푂푃 ∙ 푃퐾 2(푅2 − 푑2) 2푅2 − 푑2 푃퐾 = ⟹ 푂퐾 = 푑 푑 Page | 4 푅2푑 푂푃 2푅2 − 푑2 푃푃′ 푑2 푂퐾 ∙ 푂푃′ = 푅2 ⟹ 푂푃′ = ⟹ = ⟹ = 1 − 2푅2 − 푑2 푂푃′ 푅2 푃′푂 푅2 Property 3. P′ and 퐼(푃) = 푄 are harmonic conjugates wrt the points 푂 and 푃. Proof. 푅2 푅2 − 푑2 푃푄 푅2 − 푑2 푑2 푃푄 푃푃′ 푂푃 ∙ 푂푄 = 푅2 ⟹ 푂푄 = ⟹ 푃푄 = ⟹ = = 1 − ⟹ = 푑 푑 푂푄 푅2 푅2 푂푄 푂푃′ Remark. The converse of property 3 is also easy to prove. BARYCENTRIC COORDINATES OF THE CIRCUMCEVIAN-INVERSION PERSPECTOR OF A TRIANGLE CENTER In this section, we find the barycentric coordinates of the Circumcevian-inversion perspector of a point by using Property 2 discussed in the last section. Consider the point 푃 = (푝, 푞, 푟). Note that here, we are considering the normalized barycentric coordinates of the point. The result can be denormalized later. 2 푎 푆퐴⁄ The circumcenter 푂 has normalized barycentric coordinates ( 2푆2 , _, _) and the coordinates of 푃 are (푝, 푞, 푟). We use the barycentric distance formula to find the distance 푑 = 푃푂. Now, we use property 3. 2 2 푏 푆푏 푐 푆푐 푑2 = 푃푂2 = − ∑ 푎2 ( − 푞) ( − 푟) 2푆2 2푆2 푐푦푐 2 2 2 푏 푆푏 푐 푆푐 ′ 2 ∑ 푎 ( − 푞) ( − 푟) 푃푃 푑 푐푦푐 2푆2 2푆2 = 1 − = 1 + 푃′푂 푅2 푅2 6 2 4 2 2 2 2 2 2 2 푃′ = 8푆 푅 푝 + (4푆 푅 + ∑(푎 (푏 푆퐵 − 2푞푆 )(푐 푆퐶 − 2푟푆 )))푎 푆퐴 ∷ 푐푦푐 After algebraic manipulations and denormalization, we get 푃′ = 푝(푝 + 푞 + 푟)푎2푏2푐2 + 푎2(푏2 + 푐2 − 푎2)(푎2푞푟 + 푏2푝푟 + 푐2푝푞) ∷ It should be noted that the above expression is valid for all points 푝: 푞: 푟 instead of only for (푝, 푞, 푟) as we have denormalised the expression. SHINAGAWA COEFFICIENTS OF THE CIRCUMCEVIAN-INVERSION PERSPECTOR OF A TRIANGLE CENTER ON THE EULER LINE Suppose that 푋 is a triangle center on the Euler line given by barycentric coordinates 푓(푎, 푏, 푐) ∶ 푓(푏, 푐, 푎) ∶ 푓(푐, 푎, 푏). The Shinagawa coefficients of 푋 are the functions 퐺(푎, 푏, 푐) and 퐻(푎, 푏, 푐) such that 푓(푎, 푏, 푐) = 2 퐺(푎, 푏, 푐) ∙ 푆 + 퐻(푎, 푏, 푐) ∙ 푆퐵푆퐶. Many a times, Shinagawa coefficients of a triangle center can be conveniently expressed in terms of the following, Page | 5 2 (푆퐵 + 푆퐶)(푆퐶 + 푆퐴)(푆퐴 + 푆퐵)⁄ 푎푏푐 2 퐸 = 푆2 = ( ⁄푆) = 4푅 2 2 2 푆퐴푆퐵푆퐶⁄ (푎 + 푏 + 푐 )⁄ 2 2 퐹 = 푆2 = 2 − 4푅 = 푆휔 − 4푅 Let the Shinagawa coefficients of a triangle center 푃 on the Euler line be (푘, 푙).
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