PDF《A NOTE ON THE CENTRES OF A CLOSED》

arXiv:1804.

10345v2 [math.MG]

30 Nov

2018 A NOTE ON THE CENTRES OF A CLOSED CHAIN OF CIRCLES ? AKOS G.HORV? ATH 1. Introduction An interesting recent elementary statement on circles is Dao's Theorem on six circles (see [1], [2], [3] and [4]). This theorem states that if we have a cyclic hexagon and consider six triangles de?ned by the lines of its three consecutive sides, then the circumcentres of these triangles are the vertices of a Brianchon hexagon, that is a hexagon whose main diagonals are concurrent. Here we note that Brianchon's theorem states that the main diagonals of a hexagon circumscribed to a conic are concurrent. Hence the hexagons circumscribed to a conic are always Brianchon hexagons, see e.g. [5]). Two consecutive circles of the chain of six circles intersect each other in at most two points;

one of them is a vertex of the original hexagon and the other one we call the "second point of intersection" of the two circles. We note that the second points of intersection are not concyclic in a general situation. To see a simple example consider a degenerated Dao's con?guration (see Fig. 1). One of the sides of the hexagon has zero length and the corresponding triangle degenerates to a point I. We also assume that the circles k and l touch at a point II and similarly the circles n and p touch at a point III. The set of second points of intersection contains the points I, II and III which determine the original circumscribed circle of the pentagon. Consequently, the second points of intersection cannot be concyclic, as we stated. This shows that the following problem is independent of that of Dao. k l m n p I II III Figure 1. The set of "second points" cannot be concyclic in this Dao's con?guration. Miquel's Six-Circles Theorem (see [6]) can be formulated in the following way: If we have two cyclic quadrangle P1P2P3P4 and Q1Q2Q3Q4 for which the quadruples P1Q1Q2P2, P2Q2Q3P3, P3Q3Q4P4 are cyclic then the last quadruple of this type P4Q4Q1P1 is also cyclic. The circumcircles of the last four quadruples form a closing chain of intersecting circles with the property that the points of intersection belong to two other circles transversal to each circle of the chain. By induction, on can easily prove the following extension of Miquel's theorem: Theorem

1 ([7]). Let α and β be two circles. Let n >

2 be an even number, and take the points P1,Pn on α and Q1,Qn on β, such that each quadruple P1Q1Q2P2,Pn?1Qn?1QnPn is cyclic. Then the quadruple PnQnQ1P1 is also cyclic. In the case of n =

6 the obtained con?guration of circles is very similar to the con?guration in Dao's theorem, so it is not to surprising that L. Szilassi observed that the centers of the circles form Brianchon hexagon, but he has given no proof. G. G? evay gave a proof of this statement in [8] using projective geom- etry of the three-space. On the other hand this problem was published earlier in Crux Mathematicorum by Dao [9] without solution. It is interesting that in a later volume of Crux Mathematicorum we can ?nd Date: May, 2018.

1 2 ? A. G.HORV? ATH a correction for another problem (see [10]) containing the key statement to give a simple solution for the above one. Our short paper contains a simpler and shorter proof of that the hexagon in question is a Brianchon one. This proof also leads to a generalization of the statement from the case n =

6 to any even value of n. 2. A theorem on the chain of intersecting circles We prove the following theorem: Theorem 2. Let c(K) and c(L) be two circles with respective centers K and L. Let n >

2 be an even number, and take the points P1, . . . Pn on c(K) and Q1,Qn on c(L), such that each quadruple P1Q1Q2P2,PnQnQ1P1 is cyclic. Denote by Oi the center of the circle c(Oi) circumscribed the quadrangle PiQiQi+1Pi+1. Then for each value of i the line OiOi+1 is tangent to a ?xed conic with foci K and L. Proof. We prove that if we re?ect the point K to the successive sides of the polygon O1 . . . On then we get points on a circle c with center L. Let T denote the re?ection of K to the side O1On. From the metric de?nition of a conic it immediately follows that the locus of the re?ected image of a focus to the tangent of the conic is a circle which centre is the other focus. From this we obtained that the perpendicular bisector O1On of the segment KT is tangent to a conic with foci K and L. (See e.g. [5].) Consider the cyclic quadrangle P1Q1Q2P2 (see Fig. 2). K L O O P P Q Q T U

1 2 n O

1 1

2 2 f fgh g h Figure 2. The inscribed conic in the case of ellipse. Denote by f, g and h the re?ections in the perpendicular bisectors of Q1P1, P1P2 , P2Q2, respectively. These three lines meet at O1 (or are parallel if O1 is at in?nity), so fgh is another re?ection. Then fgh(Q2) = fg(P2) = f(P1) = Q1, so fgh is the re?ection about the perpendicular bisector of Q1Q2. But then fgh(U) = fg(K) = f(K) = T , so fgh is the re?ection about the perpendicular bisector of T U, which therefore coincides with the perpendicular bisector of Q1Q2. Consequently, the distances of the points T and U from the center L are equal to each other. Hence the conic de?ned by the foci K and L and the tangent line OnO1 agree with the conic de?ned by the same pair of foci and the tangent line O1O2. The similar reasoning for the next quadrangle P2Q2Q3P3 implies that this conic is the same as the conic de?ned by the foci K, L and the tangent line O2O3 and so on and so forth. This proves the theorem. Corollary

1 (Theorem

2 in [8]). If n =

6 the polygon de?ned by the centers Oi is a Brianchon hexagon by Brianchon's theorem on conics.

3 K U T O O O Q Q O P P O O

1 6

1 4

5 1

2 2

3 2 f g h fgh Figure 3. The inscribed conic in the case of parabola. Remark 1. Observe that in the case when K is an inner point of the circle c(L) containing the points Qi the conic is an ellipse and if the point K is an outer point of c(L) then the conic is an hyperbola. The case that the conic is a parabola occurs when the point L is "at in?nity" meaning that c(L) is a line. Indeed, in this case the segment UT is parallel to c(L) and the examined lines are tangent to a parabola (see Fig. 3). References [1] Cohl, T.: A purely synthetic proof of Daos theorem on six circumcenters associated with a cyclic hexagon, Forum Geom.

14 (2014), 261-264. [2] Dergiades, N.: Daos Theorem on Six Circumcenters associated with a Cyclic Hexagon, Forum Geom.,

14 (2014), 243-246. [3] Dung, N.T.: A simple proof of Dao's theorem on sic circles Global Journal of Advanced Research on Classical and Modern Geometries 6/1 58C61. [4] Ngo, O. D.: Some problems around the Daos theorem on six circumcenters associated with a cyclic hexagon con?gu- ration International Journal of Computer Discovered Mathematics 1/2, (2016) 40C47. [5] Glaeser, G, Stachel, H., Odehnal, B.: The Universe of Conics, SpringerSpektrum, Springer-Verlag Berlin Heidelberg, 2016. [6] Miquel, A., Theoremes sur les intersections des cercles et des spheres, Journal de mathematiques pures et appliquees Ire serie 3, (1838), 517C522. [7] G? evay, G.: Resolvable con?gurations. (accepted)

2018 [8] G? evay, G.: A remarkable theorem on eight circles. Forum Geometricorum

18 (2018), 115C118. [9] Dao, O.T.: Problem 3845, Crux Mathematicorum, 39, Issue May 2013. [10] Bataille, M. : Correction of the problem

3945 Proposed by J. Chris Fisher. Crux Mathematicorum, 41/5, May

2015 Solutions 221. ? A. G.Horv? ath, Department of Geometry, Mathematical Institute, Budapest University of Technology and Economics, H-1521 Budapest, Hungary E-mail address: ghorvath@math.bme.hu