RAZLIČITE METODE DOKAZIVANJA JEDNE TEOREME U GEOMETRIJI. (Different methods of proofs of one geometric theorem)
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1 MAT-KOL (Bnj Luk) XVII()(), 3-4 ISSN (p) ISSN (o) Metoik mtemtike RAZLIČITE METODE DOKAZIVANJA JEDNE TEOREME U GEOMETRIJI (Different methos of proofs of one geometric theorem) Šefket Arslngić i Drgoljub Milošević Sžetk: U ru je to jenest rznih okz jene teoreme iz geometrije koj se onosi n prvilni petougo. Ključne riječi: prvilni petougo, strnic i ijgonl prvilnog petougl, slični trouglovi, prlelogrm, simetrl unutršnjeg ugl trougl, rotcij, Ptolemejev i Stjurtov teorem, sinusn i kosinusn teorem, Molvjove formule, vektori, sklrni proizvo, nlitičk geoemtrij. Abstrct: In this pper we give eleven ifferent proofs of one geometric theorem for the regulr pentgon. Key wors n phrsese: regulr pentgon, sie n igonl of regulr pentgon, similr tringles, prllelogrm, ngle-bisector, rottion, Ptolemy's n Stewrt's theorem, sine n cosine lw, Mollweie's formuls, vectors, sclr prouct, nlytic geometry. AMS Subject lssifiction (): 5M4, 97G4 ZDM Subject lssifiction (): G4 U ovom ru ćemo ti jenest rznih okz jene teoreme iz geometrije koj se onosi n prvilni petougo rukovoeći se pozntom mksimom je vrijenije jen mtemtički ztk riješiti n v ili više nčin nego riješiti esetine ztk n jen te isti nčin. U ovim okzim ćemo koristiti puno činjenic iz plnimetrije, trigonometrije, vektorske lgebre, nlitičke geometrije, it. Riječ je o sljeećoj teoremi: Teorem: U prvilnom petouglu strnice i ijgonle vži jenkost =. ( ) Prirono-mtemtički fkultet Univerzitet u Srjevu, Osjek z mtemtiku, Zmj o Bosne 35, 7 Srjevo, BiH; e-mil: sefket@pmf.uns.b 7. NOU ivizije 43, 33 G. Milnovc, Srbij, e-mil: rmil47@gmil.com
2 MAT-KOL (Bnj Luk), XVII ()() Š.Arslngić i D.Milošević Dokz. Petougo ABDE je prviln, tj. AB B D DE AE (sl.). Njegov unutršnji ugo AB jenk je 8, p je u jenkokrkom trouglu AB ( AB B ): BA = AB = (8 8 ) : = 36. D E - F A Slik. B Konstruišimo už DF, F A prlelnu s strnicom EA petougl. Četverougo AFDE je romb, p je DF. Trouglovi AD i DF su jenkokrki, p je AD = AD = (8 36 ) : = 7 i DF = 8-7 = 36. Ti trouglovi su slični, što znči je A: D DF : F, tj. : : / :. q.e.. Dokz. Kko je ADF = DF (sl.), prv DF je simetrl unutršnjeg ugl AD u trouglu AD, p možemo primjeniti teoremu o simetrli unutršnjeg ugl z tj trougo: AF : F AD: D : : / : ovo je ( )., 4
3 MAT-KOL (Bnj Luk), XVII ()() Š.Arslngić i D.Milošević Dokz 3. Primjenom Ptolemejeve 3) teoreme n tetivni četverougo ADE (sl.), obijmo: ADE AED ADE / :. q.e.. Dokz 4. Rotirjmo trougo ADE oko vrh D tko se tčk E poklopi s tčkom, tčk A s tčkom F (sl.). T je ugo R AF opružen, trouglovi DF i AFD su slični. Zbog tog je: DF : D AF : AD : : / :, q.e.. E D F A B Slik Dokz 5. Koristićemo Stjurtovu 4) teoremu i primjenićemo je n trougo (sl.), p obijmo: AD 3) Ptolemeus luius, strogrčki mtemtičr, geogrf i stronom, II vijek nove ere 4) Mtthew Stewrt (77-785), škotski mtemtičr 5
4 MAT-KOL (Bnj Luk), XVII ()() Š.Arslngić i D.Milošević A DF AFF D AF AD F , p / :. q.e.. Dokz 6. Četverougo DB je prlelogrm (vii okz ), te je (sl.3): Trouglovi EB E EB B EB D EB ED D B. i A B, te ED i AD B su slični, p je: EB E DB AB AB AB. AB B ED E EB B EB AB EB AB D E B A D E A Slik 3. B 6
5 MAT-KOL (Bnj Luk), XVII ()() Š.Arslngić i D.Milošević EB Stvljjući je x AB immo x, ove x EB Immo lje zbog x AB : x x 5 x. () E EB B EB AB EB, tj. AB AB AB AB S je zbog E te E 5 5 x. () AB i AB iz (): 5, (3) 5 5. (4) 5 5 Dobijmo s iz (3) i (4): 5 5. q.e.. Dokz 7. Proužimo strnicu AB petougl o tčke F tko je AF A (sl.4). Nek je G visin jenkokrkog trougl BF; G AF. T je BG, p je AG. Immo s iz prvouglog trougl AG : cos(ag) = cos36 AG =, tj. A cos 36. () 7
6 MAT-KOL (Bnj Luk), XVII ()() Š.Arslngić i D.Milošević D E A B - - F Slik 4. Primjenom sinusne teoreme n trougo Kko je AD, obijmo: D AD, tj. sin AD sin AD sin36 sin7. () sin7 sin 36 sin36 cos 36, iz jenkosti () slijei: Iz jenkosti () i (3) immo: onosno cos 36. (3), / :. q.e.. Dokz 8. N osnovu kosinusne teoreme primjenjene n trouglove (sl.5) immo: cos36 i cos 36. BD i ABD 8
7 MAT-KOL (Bnj Luk), XVII ()() Š.Arslngić i D.Milošević D E 8 A 7 Slik B Iz prve jenkosti slijei: cos 36, p smjenom u rugu jenkost, obijmo: ove (vii okz 5): 3 3 3, (jer zbog je ) / :. q.e.. Dokz 9. Koristićemo Molvjove 5) formule: cos sin b b i, c sin c cos gje su, b,c strnice i,, unutršnji uglovi trougl AB. Primjenom prve formule n trougo BD (sl.5), obijmo: 5) Krl B. Mollweie (774-85), njemčki mtemtičr i stronom 9
8 MAT-KOL (Bnj Luk), XVII ()() Š.Arslngić i D.Milošević 8 36 cos cos36, 36 sin8 sin primjenom ruge formule n trougo ABD, immo: T je 7 36 sin sin8. 7 cos 36 cos cos36 sin8 sin8 cos 36 / :, q.e.. Dokz. Ovje ćemo koristiti vektore. Immo (npr. iz sl.): AB B A A A ( AB B) ( AB B) A AB B AB B A AB B AB B cos( AB, B) ove, zbog A, AB B i ( AB, B) 7 i činjenice je 5 cos7 obijmo 4 cos
9 MAT-KOL (Bnj Luk), XVII ()() Š.Arslngić i D.Milošević S je 5, te , q.e.. Npomen: Dćemo v okz vrijei Dokz. Immo 5 cos7. 4 sin8 sin8 cos8 sin36 cos 54 cos 38 cos8 cos8 cos8 cos8 3 4 cos 8 3cos8 4 cos sin sin 8 tj., cos8 3 4sin 8 sin8 4 sin 8 sin8 5 sin8, tj. 4 5 cos7 sin8, q.e.. 4 Dokz. Nek je z cos isin. T je n osnovu Muvreove 6) formule: z cos isin cos isin, tj z z z z z z, 6) Abrhm e Moivre ( ), engleski mtemtičr frncuskog porijekl
10 MAT-KOL (Bnj Luk), XVII ()() Š.Arslngić i D.Milošević ove zbog z : 4 3 z z z z/ : z z z z z Nek je Kko je z t ; slijei z z z. z z t t 5 t,. ( ) z cos isin z 5 5 cos i sin 5 5 cos i sin 5 5 cos i sin cos, 5 5 cos sin ove 5 5 cos z 5 z 4 (jer je cos ). Dkle, zbog cos cos7, slijei cos7, q.e.. 4 Dokz. Ovje ćemo koristiti nlitičku geometriju. Izberimo Dekrtov 7) prvougli koorintni sistem u rvni tko vrhovi A i B prvilnog petougl ABDE buu simetrični u onosu n koorintni početk O (sreište strnice AB ) i strnic AB prip pscisnoj osi (sl.6). 7) René Descrtes (596-65), frncuski mtemtičr i filozof
11 MAT-KOL (Bnj Luk), XVII ()() Š.Arslngić i D.Milošević y D(,y ) D E(-/,y ) E (/,y ) Slik 6. A(-/,) B(/,) x - Zbog AB i E (E AB ), koorinte vrhov petougl su: A,,B,,,y,D,yD i E,yE. Oreimo orintu y tčke ; immo B y, ove je y. Kko je A, A, i,, obijmo: A, ove je / :. q.e.. 3
12 MAT-KOL (Bnj Luk), XVII ()() Š.Arslngić i D.Milošević L I T E R A T U R A [] Arslngić, Š., Mtemtik z nrene, Bosnsk riječ, Srjevo, 4. [] Blgojević, V., Teoreme i zci iz plnimetrije, Zvo z užbenike i nstvn srestv, I. Srjevo,. [3] Mrić, A., Plnimetrij - zbirk rješenih ztk, Element, Zgreb, 996. Primljeno
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