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1 Introduction INTERNATIONAL JOURNAL OF GEOMETRY Vol. 9 (2020), No. 1, 5 - 24 TRIANGLES SHARING THEIR EULER CIRCLE AND CIRCUMCIRCLE PARIS PAMFILOS Abstract. In this article we study properties of triangles with given circumcircle and Euler circle. They constitute a one-parameter family of which we determine the triangles of maximal area/perimeter. We investigate in particular the case of acute-angled trian- gles and their relation to poristic families of triangles. This relation is described by an appropriate homography, whose properties are also discussed. 1 Introduction The scope of this article is to explore properties of triangles sharing their “Euler circle” ^¹#,Aº and their circumcircle ^0¹$, 2Aº. All these triangles for fixed f^, ^0g are called “admissible”, and as we’ll see shortly, they constitute a one-parameter family, which we call an “admissible family of triangles”. By well known theorems, for which a good reference is the book by Court [2], the Euler circle has its center # on the “Euler line” Y midway between the circumcenter $ and the orthocenter of the triangle (See Figure 1). A κ' κ C' Β' ε N H G O C B A' Figure 1: Euler line and circle In principle, the location of these two points, which lie symmetrically w.r. to #, can be arbitrary within certain bounds to be noticed below. Assuming the Euler circle to be con- Keywords and phrases: Triangle, Euler circle, Poristic, Triangle centers, Conics (2010)Mathematics Subject Classification: 51AO4, 51N15, 51N20, 51N25 Received: 19.10.2019. In revised form: 18.03.2020. Accepted: 02.02.2020 Triangles sharing their Euler circle and circumcircle 6 tained totally inside the circumcircle ^0¹$, 2Aº we can define corresponding admissible triangles as follows. We select an arbitrary point 0 on ^ and draw the line orthogonal to $0 inter- secting ^0 at f, g. A third point and a triangle defining an admissible triangle is found using the centroid of the triangle, lying also on line Y, its position dividing the segment $# in ratio $/# = 2. The location of the third point is on the intersection 0 \ ^0. The proof that the created triangle is admissible, is an easy exercise, which can start by observing the similarity of the triangles f$0, g which are (anti)homothetic w.r. to and in ratio 1/2. This homothety, or equivalently the parallelity of to $0, is also the criterion of choosing between and its antipode w.r. to ^0. The shape of the thus created admissible triangles depends on the relation of the ra- dius A to the distance ℎ = j#j = j#$j. For A ¡ ℎ all these triangles are acute-angled as in figure 1. For A = ℎ the triangles are right-angled as in figure 2, in which f0, g are respectively identical with f$, g and the Euler line is the median 0 of the trian- gle. Figure 3 shows the case of A < ℎ for which all admissible triangles are obtuse. The A ε κ Β' κ' C' G O N B C A' H Figure 2: The case of right-angled triangles possible locations of 0 extend not to the whole of ^ but only to the arc of the circle contained inside the circumcircle ^0. If 0 obtains the position of one of the extremities f%, &g of this arc, then the corresponding line orthogonal to $0 at 0 is tangent to ^0 and no genuine admissible triangle can be defined. The chords f%%0, &&0g of 0 0 ^ intersect ^ at two points f%0,&0g which are positions of for which the correspond- ing triangle degenerates correspondingly to the segments f%%0, &&0g. The case ℎ ≥ 3A is not possible, since then for all positions of 0 2 ^ no genuine admissible triangle can be defined. Thus, given the position of # on line Y, the point or equivalently $ has to be selected at a distance from # less than 3A. 2 Area and perimeter variation Given the Euler circle ^ and the circumcircle ^0 and considering, for the time being, the case of acute-angled triangles, we can study the variation of the area and perimeter of the admissible triangles in dependence of the variable point 0 2 ^. First we should observe that, because of the symmetry of the figure w.r. to line Y (See Figure 4), the different shapes of the admissible triangles are all obtained for the positions of 0 within the arc 12 of ^. The two extremities f1, 2g of this arc are precisely the midpoints of the corresponding sides fg of the two unique isosceli admissible triangles fg1, g2g which Triangles sharing their Euler circle and circumcircle 7 P' A ε P0 O κ' Q G B' B N Q0 H Q' κ C P Figure 3: The case of obtuse-agled triangles are symmetric w.r. to Y. A similar discussion for the circumcircle and the incircle can be found in [1]. Theorem 1. With the notation and conventions introduced so far, the two triangles fg1, g2g possess the maximal resp. minimal area and perimeter among the admissible ones. Proof. By the preceding remark, to find the maximal/minimal area/perimeter of the ad- missible triangles it suffices to examine those for which the middle 0 of their side is on the indicated arc 12. Thus, it suffices to show that the area and the perimeter are 0 decreasing as moves on this arc from 1 towards 2. For the area this can be seen by expressing it in terms of 1 = jj and the altitude D = j00j, as a function 5 ¹Gº of G = j$0j, varying in the interval corresponding to the arc 12. Latter is readily seen to be p A − ℎ ≤ G ≤ A¹A − ℎº . (1) Considering positive areas and using 5 2 ¹Gº instead, we find the expressions: 12 = 4¹4A2 − G2º, j00j = 2j$0j ¸ j00j = 2G ¸ H with GH = A2 − ℎ2 ) 2G2 ¸ A2 − ℎ2 ¹4A2 − G2º¹2G2 ¸ A2 − ℎ2º2 j00j = ) 5 2 ¹Gº = . G G2 Doing a bit of calculus we see that this is indeed strictly decreasing in the interval (1). Analogously, the decreasing behaviour of the perimeter can be seen by expressing it in terms of fD = j00j, 1 = jj,E = j000jg and doing some computation: E2 = ¹2ℎº2 − ¹H − Gº2 ) 02 = D2 ¸ ¹1/2 ¸ Eº2, 22 = D2 ¸ ¹1/2 − Eº2 ) 02 ¸ 22 = 2D2 ¸ 12/2 ¸ 2E2 = 4¹G2 ¸ 5A2 − ℎ2º , 4A¹2G2 ¸ A2 − ℎ2º 4¹¹G ¸ Aº2 − ℎ2º¹G ¸ 2Aº 0 · 2 = ) ¹0 ¸ 2º2 = ) G G r ! p p ¹G ¸ Aº2 − ℎ2 0 ¸ 1 ¸ 2 = 2 2A ¸ G 2A − G ¸ . G Triangles sharing their Euler circle and circumcircle 8 C κ B Β2 h2 2x τ2 G Β h1 1 ε H N O x B' y B'' τ1 κ' A Figure 4: Calculating the area Doing again a bit of calculus we see that the function is decreasing in the interval (1). Figure 5 gives an impression of the behavior of this perimeter function, modified by a O x x x 1 2 3 Figure 5: The perimeter function p small multiplicative factor, in the interval G1 = A − ℎ ≤ G ≤ A ¸ ℎ = G3, with G2 = A¹A − ℎº defining the right end of the interval (1). Corollary 1. The area ¹º resp. perimeter 2g of the triangle with given Euler circle ^¹#,Aº and distance from the orthocenter j#j = ℎ < A is bounded by p p ¹3A ¸ ℎº 4A2 − ¹A ¸ ℎº2 ≤ ¹º ≤ ¹3A − ℎº 4A2 − ¹A − ℎº2 , (2) p p p p p p 2¹ A − ℎ ¸ 2 Aº 3A ¸ ℎ ≤ 2g ≤ 2¹ A ¸ ℎ ¸ 2 Aº 3A − ℎ . (3) Remark 1. On the ground of a geometric definition of the ellipse, it can be proved and is well known ([16], [17, p.131]) that all admissible triangles have the following property, which we formulate as a theorem, giving also another aspect of the pair of points f$, g. Theorem 2. All triangles sharing the same Euler circle ^¹#,Aº and circumcircle ^0¹$, 2Aº, and satisfying ℎ = j#j = j$#j < A have their sides tangent to the ellipse with focals at f, $g p and eccentricity 1 − ¹ℎ/Aº2. Their corresponding side-lines are the orthogonal bisectors of the segments %, where % is a point varying on the circumcircle ^0 (See Figure 6). 3 Obtuse and right-angled trianlges Continuing with the notation and conventions of the preceding sections, the case of obtuse admissible triangles, characterized by the condition A < ℎ, is illustrated by figure 7. In Triangles sharing their Euler circle and circumcircle 9 P C κ κ' G HN O μ A B Figure 6: Enveloping ellipse of side-lines of admissible acute triangles this case the side-lines of the admissible triangles envelope a hyperbola with “auxiliary circle” ^ and “circular-directrix” ^0. By studying the variation of the chord , we see easily that the measure of the obtuse angle l varies between two values P' τ1 A ε κ' Q B δ 1 O G B' B N D B ω 2 1 Q' H ω κ 2 C P Figure 7: Enveloping hyperbola of side-lines of admissible obtuse triangles l1 ≤ l ≤ l2 . (4) The smaller l1 is the apex-angle of the maximal isosceles g1 and the bigger one is the angle l2 to which tends the measure of the obtuse angle, as the triangle tends to degen- eration to the (double) segment %%0. Notice that in this case the vertex of the variable triangle tends to %0 and the side-line tends to the tangent X of ^0 at %. Since this side-line is all the time tangent to the hyperbola, X is also tangent to it.
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