Five Proofs of an Area Characterization of Rectangles

Transcription

1 Forum Geometriorum Volume 13 (2013) FORUM GEOM ISSN Five Proofs of n re Chrteriztion of Retngles Mrtin Josefsson strt. We prove in five ifferent wys neessry n suffiient onition for onvex qurilterl to e retngle regring its re expresse in terms of its sies. There re hnful of well known hrteriztions of retngles, most of whih onerns one or ll four of the ngles of the qurilterl (see [8, p.34]). One exmple is tht prllelogrm is retngle if n only if it hs (t lest) one right ngle. Here we shll prove tht onvex qurilterl with onseutive sies,,, is retngle if n only if its re K stisfies K = 1 2 ( )( ). (1) We give five ifferent proofs of this re hrteriztion. C C D D Figure 1. Diviing qurilterl into two tringles First proof. For the re of onvex qurilterl, we hve (see the left hlf of Figure 1) K = 1 2 sin sin D 1 2 ( + ), where there is equlity if n only if = D = π 2. Using the following lgeri ientity ue to Diophntus of lexnri ( + ) 2 + ( ) 2 = ( )( ) iretly yiels the two imensionl Cuhy-Shwrz inequlity + ( )( ) Pulition Dte: Ferury 7, Communiting Eitor: Pul Yiu.

2 D 18 M. Josefsson with equlity if n only if =. Hene the re of onvex qurilterl stisfies K 2 1 ( )( ) (2) with equlity if n only if = D = π 2 n =. The thir equlity is equivlent to =, whih together with = D yiels tht tringles C n CD re similr. ut these tringles hve the sie C in ommon, so they re in ft ongruent right tringles (sine = D = π 2 ). Then the ngles t n C in the qurilterl must lso e right ngles, so CD is retngle. Conversely it is trivil, tht in retngle = D = π 2 n =. Hene there is equlity in (2) if n only if the qurilterl is retngle. C Figure 2. Congruent right tringles C n CD Seon proof. igonl n ivie onvex qurilterl into two tringles in two ifferent wys (see Figure 1). ing these four tringle res yiels tht the re K of the qurilterl stisfies 2K = 1 2 sin sin C sin D sin = 1 2 ( + )( + ) where there is equlity if n only if = = C = D = π 2. Thus K 1 4 ( + )( + ), (3) whih is known inequlity for the re of qurilterl (see [2, p.129]), with equlity if n only if it is retngle. 1 oring to the M-GM inequlity, ( + ) 2 = ( ) 1 n interesting historil remrk is tht the formul K = (this is nother re hrteriztion of retngles) ws use y the nient Egyptins to lulte the re of qurilterl, ut it s only goo pproximtion if the ngles of the qurilterl re lose to eing right ngles. In ll qurilterls ut retngles the formul gives n overestimte of the re, whih the tx olletors proly in t min.

3 Five proofs of n re hrteriztion of retngles 19 with equlity if n only if =. Similrly, ( + ) 2 2( ). Using these two inequlities in (3), whih we first rewrite, we get K 1 4 ( + ) 2 ( + ) ( ) 2( ) = 1 2 ( )( ). There is equlity if n only if =, =, n = = C = D = π 2, tht is, only when the qurilterl is retngle. n m φ Figure 3. The Vrignon prllelogrm n the imeins Thir proof. The re of onvex qurilterl is twie the re of its Vrignon prllelogrm [3, p.53]. The igonls in tht prllelogrm re the imeins m n n in the qurilterl, tht is, the line segments onneting the mipoints of opposite sies (see Figure 3). Using tht the re K of onvex qurilterl is given y one hlf the prout of its igonls n sine for the ngle etween the igonls (this ws prove in [5]), we hve tht K = mn sin φ (4) where φ is the ngle etween the imeins. In [7, p.19] we prove tht the igonls of onvex qurilterl re ongruent if n only if the imeins re perpeniulr. Hene the re of onvex qurilterl is K = mn (5) if n only if the igonls re ongruent (it is n equiigonl qurilterl). The length of the imeins in onvex qurilterl n e expresse in terms of two opposite sies n the istne v etween the mipoints of the igonls s m = 2 1 2( ) 4v 2, (6) n = 1 2 2( ) 4v 2 (see [6, p.162]). Using these expressions in (5), we hve tht the re of onvex qurilterl is given y (2( ) 4v 2 )(2( ) 4v 2 )

4 20 M. Josefsson if n only if the igonls re ongruent. Now solving the eqution 1 2 ( )( ) = 1 4 (2( ) 4v 2 )(2( ) 4v 2 ) yiels 8v 2 = 0 or = 2v 2. The seon equlity is not stisfie in ny qurilterl, sine oring to Euler s extension of the prllelogrm lw, in ll onvex qurilterls = p 2 + q 2 + 4v 2 > 2v 2 where p n q re the lengths of the igonls [1, p.126]. Thus we onlue tht v = 0 is the only vli solution. Hene onvex qurilterl hs the re given y (1) if n only if the igonls re ongruent n iset eh other. prllelogrm, the qurilterl hrterize y iseting igonls (v = 0), hs ongruent igonls if n only if it is retngle. Fourth proof. Comining equtions (4) n (6) yiels tht the re of onvex qurilterl with onseutive sies,,, is given y (2( ) 4v 2 )(2( ) 4v 2 ) sinφ where v is the the istne etween the mipoints of the igonls n φ is the ngle etween the imeins. Sine prllelogrms re hrterize y v = 0, we hve tht the re is K = 1 2 ( )( )sin φ if n only if the qurilterl is prllelogrm. In prllelogrm, φ is equl to one of the vertex ngles sine eh imein is prllel to two opposite sies (see Figure 4). prllelogrm is retngle if n only if one of the vertex ngles is right ngle. The eqution sin φ = 1 only hs one possile solution φ = π 2 ; hene we hve tht the re of onvex qurilterl is given y (1) if n only if it is retngle. φ Figure 4. The ngle etween the imeins in prllelogrm

5 Five proofs of n re hrteriztion of retngles 21 Fifth proof. onvex qurilterl with onseutive sies,,, n igonls p, q hs the re [4, p.27] 4p 2 q 2 ( ) 2. Now solving the eqution 1 2 ( )( ) = 1 4 4p 2 q 2 ( ) 2 we get (2pq) 2 ( ) 2 = 0 with only one positive solution = 2pq. Using gin Euler s extension of the prllelogrm lw = p 2 + q 2 + 4v 2, where v is the istne etween the mipoints of the igonls p n q, yiels p 2 + q 2 + 4v 2 = 2pq (2v) 2 = (p q) 2. Here the left hn sie is never negtive, wheres the right hn sie is never positive. Thus for equlity to hol, oth sies must e zero. Hene v = 0 n p = q. This is equivlent to tht the qurilterl is prllelogrm with ongruent igonls, i.e., retngle. Referenes [1] N. ltshiller-court, College Geometry, rnes & Noel, New eition y Dover Pulitions, [2] O. ottem, R. Z. Djorjević, R. R. Jnić, D. S. Mitrinović, n P. M. Vsić, Geometri Inequlities, Wolters-Noorhoff pulishing, Groningen, [3] H. S. M. Coxeter n S. L. Greitzer, Geometry revisite, Mth. sso. mer., [4] C. V. Durell n. Roson, vne Trigonometry, Dover reprint, [5] J. Hrries, re of Qurilterl, The Mthemtil Gzette 86 (2002) [6] M. Josefsson, The re of ientri qurilterl, Forum Geom. 11 (2011) [7] M. Josefsson, Chrteriztions of orthoigonl qurilterls, Forum Geom. 12 (2012) [8] Z. Usiskin n J. Griffin, The Clssifition of Qurilterls. Stuy of Definition, Informtion ge Pulishing, Chrlotte, Mrtin Josefsson: Västergtn 25, Mrkry, Sween E-mil ress: mrtin.mrkry@hotmil.om