arXiv:2104.06343v2 [math.MG] 29 Apr 2021 A note on some generalizations of Monge’s theorem Marek Lassak Abstract. We generalize Monge’s theorem for n + 1 pairwise homothetic sets (in particular convex bodies) in E n in place of three disks in E 2 . We also present a version for n + 1 independent points of E n . It also includes the reverse statement. Moreover, we give an analogon of Monge’s theorem for the n-dimensional sphere and hyperboloid model of the hyperbolic space. Mathematical Subject Classification (2010). 52A20, 52A21, 52A55. Keywords. Monge’s theorem, Menelaus’ theorem, Euclidean space, sphere, hyperbolic space 1 Introduction For any two disjoint disks in a plane, an external tangent is a line that is tangent to both disks but does not pass between them. There are two such external tangent lines for any two circles. Each such pair of external tangents for disks of different size has a unique intersection point. The classic Monge’s theorem states that for three such pairwise disjoint circles of different size the three intersection points of the external tangent lines given by the three pairs of circles always lie in a straight line. For instance, see the book [4] by Gardner. Walker [8] generalized Monge’s Theorem for n + 1 balls in the Euclidean n-space E n and a hyperplane in place of the above straight line. First we present a version of Monge’s theorem for n + 1 linearly independent points in place of the balls. This version includes also the reverse claim. Next we give a generalization of Monge’s theorem for n + 1 pairwise homothetic bounded sets (not obligatory disjoint) with homothety ratios over 1 in E n in place of the n + 1 balls in E n . A good visualization is obtained by taking convex bodies in place of our bounded sets. Moreover, we give spherical and hyperbolic n-dimensional analogons of Monge’s theorem. Our proof applies the n-dimensional Menelaus’ theorem for the n-dimensional sphere S n and hyperbolic space H n by Ushijma [7] . A question remains about possible analogs of Monge’s theorem in axiomatic geometry, where Guggenheimer [4] considers the Menelau’s theorem. 1 2 2 A note on generalizations Monge’s theorem Monge’s theorem for a wide class of sets in E n Let us start with a proposition presenting a version of Monge’s theorem for n+ 1 points in E n instead of balls. Recall that a set A of points in E n is said to be independent if the affine span of any proper subset of A is a proper subset of the span of A. Proposition 1. Let the set of points a1 , a2 , . . . , an+1 ∈ E n be independent. Consider the straight line Lij containing ai aj and a point bij ∈ Lij different from ai and aj for i, j ∈ {1, . . . , n + 1}. For every i < j denote by λij the ratio of homothety with center bij which transforms aj into ai . We −1 claim that the n(n + 1)/2 points bij belong to one hyperplane if and only if λ−1 ij λik λjk = 1 for every i, j ∈ {1, . . . , n + 1}. Proof. Let us apply the variant of Theorem 2 of the paper [1] by Buba-Brzozowa in which we take into account just lengths instead of the oriented lengths. Her n-dimensional generalization of the classic Menelaus’ theorem says that points bij , where i, j = 1, . . . , n + 1 and i < j, belong to one hyperplane of E n if and only if |ai bij | |aj bjk | |ak bik | · · = 1. |bij aj | |bjk ak | |bik ai | Since |ai bij | |bij aj | = λij for i, j ∈ {1, . . . , n + 1} and i < j, we obtain that all our points bij are in −1 one hyperplane if and only if λ−1 ij λik λjk = 1 for every i, j ∈ {1, . . . , n + 1}, which is our thesis. Clearly, if we agree that the different points a1 , . . . , an+1 in Proposition 1 are dependent, then the “if” part trivially holds true. Theorem. Assume that for sets C1 , . . . , Cn+1 ⊂ E n and for every i, j ∈ {1, . . . , n + 1} with i < j there are unique homotheties hij of ratios over 1 such that hij (Cj ) = Ci . Then the n(n + 1)/2 centers of these homotheties are in one hyperplane. Proof. Consider the n(n + 1)/2 homotheties hij such that hij (Cj ) = Ci , where i, j ∈ {1, . . . , n + 1} and i < j (see Figure for the special case of three convex bodies in E 2 ). Of course, for every three −1 homotheties hpq , hpr , hqr , where p < q < r, we have hpr (h−1 qr (hpq (Cp ))) = Cp . By the assumed uniqueness of the homotheties hij there are points o1 , . . . , on+1 such that hij (oj ) = oi for every i, j ∈ {1, . . . , n + 1} with i < j. From the “if” part of Proposition 1 we get our thesis. Marek Lassak 3 Figure. Illustration of Theorem for the case of three convex bodies in E 2 If C1 , . . . , Cn+1 from Theorem are centrally-symmetric convex bodies, we may rephrase this as follows: Monge’s theorem holds for n + 1 balls of different sizes of a normed n-dimensional space in place of n + 1 balls of Euclidean space. A particular case is for the two-dimensional Lp spaces considered in the preprint [2] by Ermiş and Gelişgen. The author thanks them for sharing their preprint [2], which was mobilizing to show the above theorem. The assumption that homotheties hij are unique holds always if the sets are bounded and non-empty. It also holds for some unbounded sets. For instance when in E 2 we take Ci as the intersection of half-planes x ≥ 0, y ≥ 1/i and x + y ≥ 4 − i, where i = 1, 2, 3. Then all bij are different and lie on the line x = 0. If we exchange y ≥ 1/i into y ≥ 0 here, all bij coincide and still are on x = 0. For example the assumption does not hold for any family of n + 1 translates of a half-space in E n ; the thesis may be not true for some homotheties with ratios over 1 between them. We let the reader to show that if the latter assumption does not hold, then there are some positive homotheties for which the thesis of Theorem is still true. 3 Analogons of Monge’s theorem in spherical and hyperbolic spaces Below by X n we denote both the n-dimensional sphere S n and the hyperboloid model H n of the hyperbolic n-dimensional space. By a hyperplane and a line of X n we mean a subset of X n isometric to X n−1 and X 1 , respectively. Usually, for S n they are called a (n − 1)-dimensional subsphere and a great circle, respectively. By the distance |xy| of points x, y ∈ X n (which are not opposite for X n = S n ) we mean the length of the geodesic joining x and y. By the arc xz between x and z we mean the set of points y such that |xy| + |yz| = |xz|. Let c, p, r ∈ X n be points such that p ∈ cr 4 A note on generalizations Monge’s theorem and c 6= p, or r ∈ cp and c 6= r. If |cr| = λ · |cp|, then we say that r is the image of p under the X n -homothety with center c and ratio λ. Clearly λ > 0. We call a set A ⊂ X n (embedded in E n+1 ) to be independent if the set A ∪ {o}, where o is the origin of E n+1 , is independent in E n+1 . The Menelau’s theorem on S 2 is recalled in Proposition 66 of the book [5] by Rashed and Papadopoulos and its variant for H 2 is presented by Smarandache and Barbu [6]. Recently their generalizations for S n and H n are giver in Theorem 4 of Ushijima [7]. From this result, similarly to the proof of our Proposition 1 for E n , we get the following Proposition 2 on the analogous variant of Monge’s theorem for n + 1 points on S n and H n . Here by λij we mean sin |ai bij | sin |bij aj | for S n and sinh |ai bij | sinh |bij aj | for H n . Proposition 2. Let a1 , a2 . . . , an+1 be a set of independent points of X n . Consider the line Lij containing ai aj . Denote by bij a point different from ai and aj in Lij such that aj ∈ ai bij for i, j ∈ {1, . . . n + 1}. For every i < j denote by λij the ratio of the X n -homothety with center bij such that ai is the image of aj . Then the n(n + 1)/2 points bij are in one hyperplane of H n if and only if −1 λ−1 ij λik λjk = 1 for every i, j ∈ {1, . . . , n + 1}. References [1] M. Buba-Brzozowa, Ceva’s and Menelaus’ theorems for n-dimensional space, J. of Geom. Graph. 4 (2000), No. 2, No. 115–118. [2] T. Ermiş and Ö. Gelişgen, Does the Monge theorem apply to some non-Euclidean geometries?, arXiv:2104.04274v1. [3] M. Gardner, The Collosal Book of Short Puzzles and Problems, W. W. Norton & Company, 2006. [4] H. Guggenheimer, The theorem of Menelaus in axiomatic geometry, Geom. Dedicata 3 (1974), 257–261. [5] R. Rashed and A. Papadopoulos, Menelau’s Spherics, Early translation and al-M Żah Żan Żi /al-Haraw Żi’s version. Dual Arabic-English text, De Gruyter, Berlin 2017. [6] F. Smarandache and C. Barbu, The hyperbolic Menelaus theorem in the Poincare’ disc model of hyperbolic model geometry, Ital. J. Pure Appl. Math., 30 (2013) 67–72. [7] A. Ushijima, Ceva’s and Menelaus’ theorems for higher-dimensional simplexes, J. of Geom. 110 (2019), no. 13, 9p. [8] R. Walker, Monge’s theorem in many dimensions, Math. Gaz. 60 (1976), 185–188. Marek Lassak University of Science and Technology Bydgoszcz 85-798, al. Kaliskiego 7, Poland e-mail: lassak@utp.edu.pl