Applied Mathematical Sciences, Vol. 7, 2013, no. 32, 1557 - 1561 HIKAR - PDF document

Applied Mathematical Sciences, Vol. 7, 2013, no. 32, 1557 - 1561 HIKARI Ltd, www.m-hikari.com Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University, Pathumthani 12121 Thailand banyat@mathstat.sci.tu.ac.th Copyright 2013 Banyat Sroysang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this paper, we show that the regular heptagon of diameter one is the smallest regular heptagon which contains all triangles of perimeter Keywords: Cover, Heptagon, Triangle, Worm Problem 1 Introduction and Preliminaries for a family of arcs is a convex region containing a congruent copy of every arc in the family. Wetzel [5-6] determined both the smallest equilateral triangular cover and the smallest rectangular cover for the family of all triangles perimeter two. In 2000, Furedi and Wetzel [1] determined the smallest convex cover for the family of all triangles ofperimeter two. In 2009, Zhang and Yuan [7] determined the smallest regularized parallelogram cover for the family of all triangles ofperimeter two. In 2011, Sroysang [3,4] determined the smallest regularized trapezoid cover for the family of all triangles ofperimeter two. In this paper, we show that the regular heptagon of diameter one is the smallest regular iangles of perimeter two. The maximum and the minimum of the support lines of the set are called the of the set and the thickness of , respectively. We note that the diameter and the thickness of any triangle are the length of the longest side and the length of the altitude to the longest side, perimeter two is at least 23, 1558 B. Sroysangand the thickness of any triangle ofperimeter two is at most . Moreover, the diameter of any regular heptagon is the length of the diagonal. Thus, the diameter of any regular heptagon cover for the family of all triangles ofperimeter two is at least 1. 2 Results The smallest regular heptagon cover for the family of all triangles ofperimeter two is the regular heptagon of diameter onethe area is Let be the regular heptagon of diameter one and let be a triangle ofperimeter two with the vertices and such that the angle is greater than or equal to the angle is greater than or equal to the angle be a vertex of the regular heptagon and be two points on the perimeter of the regular heptagon such that the distance between the and the segment is equal to 0.6 and the segment is parallel to a diagonal as shown in Fig. 1. We note that || into two cases. Case 1. The diameter of the triangle is at most ||. We note that the distance between the and the is greater than || as shown in Fig. 1. WLOG, we can put the triangle into the regular where the segment lies on the segment implies that the regular heptagon Fig. 1. The segment in the regular heptagon Case 2. The diameter of the triangle is greater than ||. 1560 B. Sroysang in the triangle Define ()1tan(1)tanLxxxxx§·§·=−++−+¨¸¨¸©¹©¹ is the total length of the perimeter of the right triangle . By the calculation on , we obtain that ()(0)2LxL as shown in Fig. 5. 0.02 .04 .06 0.1 2.05 2.1 2.15 2.2 2.25 2.3 where 00.1 Since the total length of the perimeter of the triangle is greater than the total length of the perimeter of the right triangle , it follows that the total length of the perimeter of the triangle is greater than two. This is a contradiction. Hence, the regular heptagon is a cover for the family of all triangles ofperimeter two. r the family of all triangles ofperimeter two must cover the line segment of length one. Hence, the diagonal of any regular heptagon cover for the family of all triangles ofperimeter two must have length at has length one. Therefore, the Triangles in regular heptagons 1561 is a smallest cover for the family of all triangles ofperimeter Acknowledgement. This research was supported by the Thammasat University This research was supported by the Thammasat University Z. Furedi and J. E. Wetzel, The smallest convex cover for triangles of perimeter two, Geom. Dedicata, dicata, L. Moser, Poorly formulated unsolved problems of combinatorial geometry, Mimeographed (1996). ographed (1996). B. Sroysang, Regularized Trapezoid Cover for Isoperimetric Triangles, Int. J. Comput. Appl. Math., put. Appl. Math., B. Sroysang, Right Trapezoid cover for triangles of perimeter two, Kasetsart tsart J. E. Wetzel, Boxes for isoperimetric triangles, Math. Mag., (2000), (2000), J. E. Wetzel, The smallest equilateral cover for triangles of perimeter two, ter two, Y. Zhang and L. Yuan, Parallelogram cover for triangles of perimeter two, Southeast Asian Bull. Math., Received: January, 2013