ABOUT A CONSTRUCTION PROBLEM

Transcription

1 INTERNATIONAL JOURNAL OF GEOMETRY Vol 3 (014), No, ABOUT A CONSTRUCTION PROBLEM OVIDIU T POP ad SÁNDOR N KISS Abstract I this paper, we study the costructio of a polygo if we kow the midpoits of their sides 1 Itroductio I this paper, we search the aswer to the ext problem: i a plae there are give disticts poits, N, 3; let s built a polygo such that the midpoits of the polygo sides to be give poits I the secod sectio of paper, we will demostrate that this costructio is possible, ad for the odd values of, it is also uique For the realizatio of our purpose, we use Varigo s Theorem ad we will idetify, i some cases, the poit with their afixe As demostratio, we will use the mathematical iductio I the third sectio, we will demostrate how the cetroid of a polygo could be built ad we will characterized the cetroid of a polygo with the afixes of their vertexes Also, we will give a property which give a sufficiet coditio as the border of polygo to be closed if the polygo is costructed from the midpoits The followig result is well-kow Theorem 11 (Varigo s Theorem, 1731) Let ABCD be a quadrilateral If M, N, P, Q are the midpoits of the sides AB, BC, CD, respectively DA, the MNP Q is a parallelogram ad T [MNP Q] = T [ABCD], where T [ABCD] is the area of quadrilateral ABCD The costructio of a polygo whe we kow the midpoits of their sides We are goig to study the ext problem: givig the midpoits of a polygo, costruct the polygo Propositio 1 Givig three o colliear poits, there exists a uique triagle with the property that the give poits are also the midpoits of the triagle s sides 010 Mathematics Subject Classificatio 51M04, 51M15, 51M30 Key words ad phrases Cetroid for polygos Received: I revised form: Accepted:

2 About a costructio problem 15 Proof Let M, N, P be the give poits We costruct from P the parallel to MN, from M the parallel to NP ad from N the parallel to MP (Figure 1) This parallels cross each other two by two, obtaiig the seeked triagle ABC This triagle was obtaied i a uique way A M N B P C Figure 1 Propositio Givig o colliear poits so that MNP Q is a parallelogram ad cosiderig a arbitrary poit A, there exist B, C, D so that M, N, P, Q are midpoits of sides AB, BC, CD, respectively DA Proof We ote by z A the afixe of poit A Because MNP Q is a parallelogram, we have (1) z M + z P = z N + z Q By takig ito accout that M, N, P, Q are midpoits (Figure ), we have that z B = z M z A, z C = z N z B = z N z M + z A, z D = z P z C = z P z N + z M z A From (1), we obtai that z D = z Q z A ad the z A + z D = z Q, so Q is the midpoit of side AD B(z B ) M(z M ) A(z A ) N(z N ) Q(z Q ) C(z C ) D(z D ) P(z P ) Figure Propositio 3 Givig five poits, there exists a uique petago with the property that the give poits are the midpoits of the petago s sides

3 16 Ovidiu T Pop ad Sádor N Kiss Proof We ote by M, N, P, Q, S these five poits (Figure 3) A N C S E M Q T Figure 3 B P D We costruct the poit T so that SQP T is a parallelogram Now, there exists a uique triagle ABC, so that the midpoits of its sides are also the poits M, N, T By takig Propositio ito accout, startig from B, we obtai the poits D ad E The seeked petago is ABDEC Further, by usig the idea from [4], we will exted the results demostrated i the propositios above Theorem 1 Let N ad + 1 be give poits The, there exists a uique polygo with +1 sides such that, the give poits are the midpoits of the sides Proof We prove this theorem by mathematical iductio For = 1 ad =, by takig Propositio 1 ad Propositio 3 ito accout, the coclusios of Theorem 1 take place Let k N, ad we assume that the coclusios of Theorem 1 take place for m N, m k We prove that the coclusio of Theorem 1 is also true for k + 1 Let M 1, M,, M k+3 be k + 3 poits (Figure 4) M 1 M k+3 A 3 M Ṃ k M k k+3 M k+ k+ M k+1 Figure 4 With the poits M k+1, M k+, M k+3, there exists a uique poit M so that the quadrilateral M k+1 M k+ M k+3 M is a parallelogram Now, we have the poits M 1, M,, M k, M, so accordig to the hypothesis of iductio, there exists a uique polygo k+1 so that the poits M 1, M,, M k, M are the midpoits of the sides, A 3,, k k+1, k+1 With the help of the poits k+1 ad, by takig Propositio ito accout, the poits k+ ad k+3 are determied k+1

4 About a costructio problem 17 3 The cetroid of a polygo I the followig, usig the idea from [4], we will give the afixe of the cetroid for a polygo Let A 3 be a triagle, M 1, M, M 3 the midpoits of the sides A 3, A 3, respectively The M 1, M, A 3 M 3 are called medias ad are cocurret lies i a poit G, called the cetroid (Figure 31) ad we have M 3 G M 1 Figure 31 M A 3 (31) z G = z + z A + z A3 3 ad (3) where z G is the afixe of poit G G GM 1 = G GM = GA 3 GM 3 = 1, Let A 3 A 4 be a quadrilateral ad G 1, G, G 3, G 4 the cetroids for the triagles A 3 A 4, A 3 A 4, A 4, respectively A 3 The segmets G 1, G, A 3 G 3, A 4 G 4 are called medias (Figure 3) ad are cocurret i a poit G, called the cetroid of the quadrilateral (see [3]) ad we have A 4 M 3 G G G1 Figure 3 A 3 (33) z G = z M 1 + z M + z M3 + z M4 4 ad G (34) = G = GA 3 = GA 4 = 3 GG 1 GG GG 3 GG 4 1 We assume for every defied k 1 the otio of media ad cetroid for polygo of k sides, so the media is a segmet determied by a vertice of the give polygo ad a cetroid determied by other k 1 sides of the give polygo These medias are cocurret i a poit called cetroid of the give polygo, ad this poit divides each media i the rate k 1 1 We cosider the polygo A with sides, ad let G 1, G,, G be the cetroid of the polygos A 3 A, A 3 A,, respectively A 1 (Figure 33) Let S be the cetroid of polygo A The, the medias G 1, G,, A G are cocurret i a poit G, called the cetroid of polygo

5 18 Ovidiu T Pop ad Sádor N Kiss A ad the, we have (35) z G = z + z A + + z A ad GA (36) = GA 1 = = G = 1 GG GG 1 GG 1 1 A A -1 G A - G -1 S Figure 33 Theorem 31 Let N, ad be a polygo, M 1, M,, M are the midpoits of the sides, A 3,, respectively The the polygos M 1 M 3 M 1, M M 4 M ad have the same cetroid Proof Takig (35) ito accout, we have z G1 = z M 1 + z M3 + + z M 1, where G 1 is the cetroid of the polygo M 1 M 3 M 1 But M 1 is the midpoit of the side, so z M1 = z + z A z M3 = z A 3 + z A4 from where z G1 = z + z A + + z A G A -3, ad similar,, z M 1 = z 1 + z A If G is the cetroid of the polygo M M 4 M, similar we obtai that z G = z + z A + + z A The cetroid of the polygo have the afixe z G = z + z A + + z A From the remarks above, the coclusio of Theorem 31 follows Theorem 3 Let N, ad the give poits M 1, M,, M For let be the symmetric to a poit M 1, A 3 the symmetric to a poit M,, A 1 the symmetric to a poit M The, for ay from the pla, coicide with A 1 if ad oly if the polygos M 1M 3 M 1 ad M M 4 M have the same cetroid,

6 Proof We have About a costructio problem 19 z A = z M1 z A1, z A3 = z M z A = z M z M1 + z A1, z A4 = z M3 z A3 = z M3 z M + z M1 z A1,, z A = z M 1 z M + z M 3 + z M1 z A1, so z A 1 = z M z A = z M z M 1 + z M z M1 + z A1 The ad A 1 coicide if ad oly if z M z M 1 + z M z M z M z M1 + z A1 = z A1, equivalet to z M + z M + + z M = z M 1 + z z M1, equivalet to z M1 + z M3 + + z M 1 = z M + z M4 + + z M, equivalet with the polygos M 1 M M 1 ad M M 4 M have the same cetroid Refereces [1] Adrica, T, Metric relatios i the plae obtaied by usig complex umbers, Creative Math & If 19 (010), No, [] Barbu, C, Fudametal Theorems of Triagle Geometry, Ed Uique, Bacău, 008 (Romaia) [3] Efremov, D, The ew geometry of the triagle, Biblioteca Olimpiadelor de Matematică 4, Ed Gil, 010 [4] Golovia, L I ad Iaglom, I M, Iductio i Geometry, Ed Tehică, Bucharest, 1964 (Romaia) [5] Pop, O T, Miculete, N ad Becze, M, A Itroductio to Quadrilateral Geometry, Ed Did Ped, Bucharest, 013 [6] Szöllősy, Gh ad Pop, O T, A ew proof of Neuberg s Theorem ad oe applicatio, Iteratioal Joural of Geometry, 1 (01), No 1, 5 9 NATIONAL COLLEGE MIHAI EMINESCU 5 MIHAI EMINESCU Street, SATU MARE , ROMANIA ovidiutiberiu@yahoocom CONSTANTIN BRÂNCUŞI TECHNOLOGY LYCEUM SATU MARE, ROMANIA dsadorkiss@gmailcom