Right Trapezoid Cover for Triangles of Perimeter Two

Size: px

Start display at page:

Download "Right Trapezoid Cover for Triangles of Perimeter Two"

Trevor Griffith
6 years ago
Views:

1 Kasetsart J (Nat Sci) 45 : 75-7 (0) Right Trapezoid Cover for Triangles of Perimeter Two Banyat Sroysang ABSTRACT A convex region covers a family of arcs if it contains a congrent copy of every arc in the family In 000, John E Wetzel determined the smallest rectangle which covers the family of all triangles with perimeter two In this paper, we search for the smallest right trapezoid which covers the family of all triangles of perimeter two Keywords: worm problem, cover, triangle, perimeter, trapezoid INTRODUCTION Moser (9) posed the qestion, the worm problem, of what is the region of smallest area which contains a congrent copy of every nit arc on the plane For a convex region R and a family F of arcs, we say that R covers F or R is a cover for F, if R contains a congrent copy of each arc in F A poplar shape of arcs is a triangle Conveniently, we do not consider the problem abot the family of all triangles with perimeter one bt we consider the problem abot the family of all triangles with perimeter two However, we can scale between triangles with perimeter one and triangles with perimeter two Wetzel (997) determined the smallest eqilateral trianglar cover for the family of all triangles with perimeter two Wetzel (000) determined the smallest rectanglar cover for the family of all triangles with perimeter two Fredi and Wetzel (000) determined the smallest convex cover for the family of all triangles with perimeter two Zhang and Yan (009) determined the smallest reglarized parallelogram (whose length of the smaller diagonal is not less than one) cover for the family of all triangles with perimeter two; this reslt implies the reslt in Wetzel (000) In this paper, we present the smallest reglarized right trapezoid (whose length of the smaller diagonal is not less than one) cover for the family of all triangles with perimeter two This new reslt also implies the reslt in Wetzel (000) MATERIALS AND METHODS In this section, we list the basic definitions and a basic reslt Definition A right trapezoid is called a reglarized right trapezoid if the length of its smaller diagonal is not less than one Definition A right trapezoid R is circmscribed abot a triangle T, if T is a sbset of R and each side of R contains a vertex of T Conveniently, we let R(,, v) be the right trapezoid with the smaller angle and the Department of Mathematics and Statistics, Faclty of Science and Technology, Thammasat University, Pathm Thani, Thailand banyat@mathstatscitacth Received date : /0/ Accepted date : 8/04/

2 Kasetsart J (Nat Sci) 45(4) 757 lengths, v ( v) of two adjacent sides of the angle Lemma For each triangle T with perimeter two, if R(,, v) is a circmscribed right trapezoid of T as shown in Figre, then + v vcos Proof Since each side of any triangle with perimeter two has length less than one, it is clear if Figre (a) occrs Assme that R(,, v) is a circmscribed right trapezoid of a triangle T with perimeter two, as shown in Figre (b) We constrct a congrent copy of both R(,, v) and T by a 80 rotation, and label all vertices as shown in Figre Since the angle EHG is at least 90, we obtain that BC BC Then FH FF FB + BC + C F FB + BC + CF FB + BC + CF Ths, + v vcos FH Definition Let X be a convex set in the plane For each θ [0, π], let ω X (θ) be the distance between the two parallel spport lines of X with angle of inclination θ The minimm of ω X (θ) is called the thickness of X and the maximm of ω X (θ) is called the diameter of X For each convex set X, we know that ω X (θ) is a continos fnction of θ For each triangle T with perimeter two, the thickness of T is the length of the altitde to the longest side; the diameter of T is the length of the longest side Since the eqilateral triangle with perimeter two has altitde, it follows that, for each triangle T with perimeter two, the thickness of T is at most and the diameter of T is at least RESULTS AND DISCUSSION In this section, we present the main reslts abot the smallest reglarized right trapezoid cover for the family of all triangles with perimeter two Theorem Let (0, 90 ] Then R(, csc, + cot ) is the smallest reglarized right trapezoid cover with angle for the family of all triangles with perimeter two, and the least area is +cot Proof First, we will show that v F' E' (a) v G' B' C C' H B G (b) E F Figre Two circmscribed right trapezoids Figre A 80 -rotation of Figre (b)

3 758 Kasetsart J (Nat Sci) 45(4) R(, csc + cot, ) is a cover for the family of all triangles with perimeter two Let T be a triangle with perimeter two Then there are angles θ,θ [0, 0 ] sch that the thickness and the diameter of T are ω T (θ ) and ω T (θ ), respectively Then ω T (θ ) and ω T (θ ) Since ω T is continos, by the intermediate vale theorem, there is an angle θ [θ, θ ] sch that ω T (θ) Let L and L be the two parallel spport lines of T with angle of inclination θ Then L contains a vertex of T and L contains a vertex of T Let A, B and C be vertices of T Under the rigid motions of T, withot loss of generality, we can assme that T pt on the position as shown in Figre ; the angle α is at most 90 Let D be a vertex in L sch that the segment CD is perpendiclar to L Then CD If 80 - BAD, then we throgh B draw the segment EF with CFE as shown in Figre (a); otherwise, we throgh A draw the segment EF with CFE as shown in Figre (b) Then the right trapezoid CDEF covers T Note that EF csc By Lemma, we have CE CF + EF - CF EF cos Then CF cot < cot + CE + + cot This implies that R(, csc + cot, ) also covers T Next, we sppose that a reglarized right trapezoid R 0 (,, v) is a cover for the family of all triangles with perimeter two bt R 0 (,, v) is not a congrent copy of R(, + cot ) Then csc, + v vcos Since R 0 (,, v) covers the eqilateral triangle with perimeter two, we have the length csc θ θ Case csc and + v vcos > ( ) ( ) Then + v v cos sin > E B F AE F A α B ( sin ) α (a) (b) D D C C L L L L Figre The right trapezoid CDEF is a circmscribed right trapezoid of the triangle ABC

4 Kasetsart J (Nat Sci) 45(4) 759 Then the length v > cos + Then the area is ( v cos )( sin ) > cos + ( sin ) cos + +cot Ths, the area of R(, csc + cot, ) is less than the area R 0 (,, v) Case > csc and + v vcos Then v( cos ) ( v) Then v ( cos ) Then the area is ( v cos )( sin ) ( v v cos )( sin ) cos v sin cos sin ( cos ) > +cot, de to their derivatives on (0, 90 ] Ths, the area of R(, csc + cot, ) is less than the area R 0 (,, v) Case > csc and + v vcos > We scale R 0 (,, v) to a smaller similar reglarized right trapezoid R (, x, y), for which at least one condition holds: (i) x csc and x + y xycos, (ii) x csc and x + y xycos Then the area of R(, csc + cot, ) is less than or eqal to the area R (, x, y) Ths, the area of R(, csc + cot, ) is less than the area R 0 (,, v) Therefore, R(, csc, + cot ) is the smallest reglarized right trapezoid cover with angle for the family of all triangles with perimeter two Corollary (Wetzel, 000) The rectangle R(90,, ) is the smallest rectangle cover for the family of all triangles with perimeter two, and the area is Proof This follows Theorem in case 90 Theorem Let (0, 90 ] Assme that R 0 (, 0, v 0 ) is a reglarized right trapezoid Then, the smallest reglarized right trapezoid R (,, v) similar to R 0 (, 0, v 0 ) that covers the family of all triangles with perimeter two is determined as follows: (a) if v 0 cos + sin, then the length csc and the length v ; 0

5 70 Kasetsart J (Nat Sci) 45(4) (b) if v 0 < cos + sin, then the length and + cos the length v + v v cos Proof (a) Assme that cos + sin v Then v 0 csc ( cos + sin ) + cot Then R (,, v) contains a congrent copy of R(, csc + cot, ) Ths R (,, v) is a cover for the family of all triangles with perimeter two Since the eqilateral triangle with perimeter two mst be covered, there is no smaller reglarized right trapezoid cover This proves (a) (b) Assme that < cos + sin Then (v 0-0 cos ) ( 0 sin ) Then 0 + v 0 v 0 0 cos 0 + ( 0 sin ) ( 0 cos ) 0 + ( 0 sin ) 0 ( 0 sin ) Then v 0 + cos ( sin ) 0 csc Moreover, + v vcos Then v cos + ( sin ) + v v cos and no smaller reglarized right trapezoid similar to that covers the family of all triangles with perimeter two Next, we will show that R (,, v) is a cover for the family of all triangles with perimeter two Let T be a triangle ABC with perimeter two and longest side a BC Note that a + v vcos Case sin a Then there is a direction θ sch that ω R (θ) sin Similar to the proof of theorem, the reglarized right trapezoid R (,, v) covers T Case sin > a Then sin h where h is the distance between A and BC Ths v sin max{a, h} It is easy to see that the reglarized right trapezoid R (,, v) covers T This proves (b) Corollary (Wetzel, 000) Let R 0 (90, 0, v 0 ) be a reglarized right trapezoid Then, the smallest reglarized right trapezoid R (90,, v) similar to R 0 (90, 0, v 0 ) that covers the family of all triangles with perimeter two has sides: < 0 (a) (b) and if v 0 0 ; and if v Proof This follows Theorem in case 90

6 Kasetsart J (Nat Sci) 45(4) 7 CONCLUSION We smmarize, for (0,90 ], that the smallest reglarized right trapezoid cover for the family of all triangles with perimeter two has area +cot This reslt implies the reslt in Wetzel (000) abot the rectanglar cover for the family of all triangles with perimeter two ACKNOWLEDGEMENTS This research was spported by the Thammasat University Research Fnd 00, Thailand LITERATURE CITED Fredi, Z and JE Wetzel 000 The smallest convex cover for triangles of perimeter two Geom Dedicata 8: 85 9 Moser, L 9 Poorly Formlated Unsolved Problems of Combinatorial Geometry Mimeographed (Reprinted in Discrete Appl Math : 0 5) Wetzel, JE 997 The smallest eqilateral cover for triangles of perimeter two Math Mag 70: 5 0 Wetzel, JE 000 Boxes for isoperimetric triangles Math Mag 7: 5 9 Zhang, Y and L Yan 009 Parallelogram cover for triangles of perimeter two Sotheast Asian Bll Math : 07

Cubic graphs have bounded slope parameter

Cubic graphs have bounded slope parameter Cbic graphs have bonded slope parameter B. Keszegh, J. Pach, D. Pálvölgyi, and G. Tóth Agst 25, 2009 Abstract We show that every finite connected graph G with maximm degree three and with at least one