A remark on Chapple-Euler Theorem in non Euclidean geometry
aa r X i v : . [ m a t h . M G ] D ec A remark on Chapple-Euler Theorem in nonEuclidean geometry
Takeo Noda and Shin-ichi YasutomiJanuary 6, 2021
Abstract
In non-Euclidean geometry, there are several known correspond-ings to Chapple-Euler Theorem. This remark shows that those re-sults yield expressions corredponding to the well-known formula d = p R ( R − r ). For a triangle, let R , r and d be its circumradius, inradius and the distancebetween its circumcenter and its incenter respectively. In 1746, Chapple gavefollowing Theorem in Euclidean geometry. Theorem 1.1 ([3]) . d = p R ( R − r ) . In 1765, Euler (See [2]) also reached the same result, which is calledChapple-Euler Theorem. Theorem 1.1 implies an inequality R ≥ r , whichis called Euler inequality or Chapple-Euler inequality.We consider spherical geometry by a surface of constant curvature K = 1and hyperbolic geometry with curvature K = −
1. In spherical geometry
Mathematics Subject Classification . Primary 51M09;
Theorem 1.2 ([4]) . Given a triangle in spherical, or hyperbolic geometryfollowing formula holds respectively. sin d = sin ( R − r ) − sin r cos R ( spehrical ) , sinh d = sinh ( R − r ) − sinh r cosh R ( hyperbolic ) . Alabdullatif also showed the following formula in hyperbolic geometry.
Theorem 1.3 ([1]) . tanh r = tanh( R + d )(cosh R (sinh r −
1) + cosh ( r + d ))cosh ( r + d ) − cosh R cosh r . In this article, we give the following Euler-Chapple type formulas in spher-ical and hyperbolic geometry, which are equivalent to the above theorems.
Theorem 1.4.
Given a triangle in spherical, or hyperbolic geometry follow-ing formula holds for in spherical and hyperbolic geometry, respectively. tan d = p tan R (tan R − r ) p tan r + (1 + tan r tan R ) , if R < π , , if R = π , ( spherical ) , tanh d = p tanh R (tanh R − r ) q − tanh r + (1 − tanh r tanh R ) ( hyperbolic ) . This theorem implies the following Euler-Chapple type inequalities inspherical and hyperbolic geometry:
Corollary 1.5.
The circumradius R and inradius r of a triangle in sphericalor hyperbolic geometry satisfy the following inequality, respectively: tan R ≥ r ( spherical ) , tanh R ≥ r ( hyperbolic ) . Remark . These inequalities have been shown directly by Svrtan and Vel-jan [5]. 2 roof of Theorem 1.4 .First, we consider a triangle in spherical geometry. In the case R = π ,both of circumscribed and inscribed circles coincide to a great circle, so d = 0or d = π and therefore tan d = 0.Thus we may assume r < R < π in the following. By Theorem 1.2, wehave sin d = sin ( R − r ) − sin r cos R. Hence, we havetan d d = tan ( R − r )1 + tan ( R − r ) − tan r r
11 + tan R , which impliestan d (tan r + (1 + tan r tan R ) ) = tan R (tan R − r ) . (1.1)From (1.1) and the facts that 0 ≤ d < π and 0 < r < π , we have Theorem.Next, we consider a triangle in hyperbolic geometry. By Theorem 1.2, wehave sinh d = sinh ( R − r ) − sinh r cosh R. Similarly, we havetanh d ( − tanh r + (1 − tanh r tanh R ) ) = tanh R (tanh R − r ) . (1.2)First, we suppose − tanh r + (1 − tanh r tanh R ) = 0 . (1.3)Then, from (1.2) we have tanh R − r = 0 . (1.4)From (1.3) and (1.4) we have tanh R = 1 , − , , −
2, which contradicts 0 < tanh R <
1. Next, we suppose − tanh r + (1 − tanh r tanh R ) < . (1.5)3y making the triangle continuously smaller and smaller, we see that R and r are sufficiently small so that − tanh r + (1 − tanh r tanh R ) >
0. Hence,there exists a triangle such that − tanh r + (1 − tanh r tanh R ) = 0, whichcontradicts the previous consideration. Therefore, we have − tanh r + (1 − tanh r tanh R ) > . (1.6)From (1.2) and (1.6) we have Theorem. We note that we can also prove thehyperbolic case from Theorem 1.3. (cid:3) References [1] A. Alabdullatif; Hyperbolic variants of Poncelet’s theorem, Thesis, Uni-versity of Southampton, Doctoral Thesis (2016).[2] O.Bottema et al ; Geometric Inequalities, Wolters-noordhoff Publishing,151p (1969).[3] W.Chapple; an essay of the properties of triangles inscribed in andcircum- scribed about two given circles, Miscellanea curiosa mathemat-ica 4, (1746), 117-124.[4] K.Cho and J.Naranjo; Extensions of Euclidean Relations and Inequali-ties to Spherical and Hyperbolic Geometry, REU at Oregon State Uni-versity, (2017).[5] D.Svrtan and D.Veljan; Non-Euclidean Versions of Some Classical Tri-angle Inequal-ities.Forum Geometricorum, Vol. 12, (2012), 197-209.Takeo Noda: Faculty of Science, Toho University, JAPAN
E-mail address: [email protected]
Shin-ichi Yasutomi: Faculty of Science, Toho University, JAPAN