Sličnost (geometrija) — разлика између измена

Садржај обрисан Садржај додат

Инлајн

Верзија на датум 27. јун 2023. у 20:09

Figure koje su prikazane istom bojom su slične

Dva geometrijska objekta se nazivaju sličnim ako imaju isti oblik, ili jedan ima isti oblik kao lik u ogledalu drugog. Preciznije, jedan se može dobiti od drugog ravnomernim skaliranjem (uvećavanjem ili smanjivanjem), i po potrebi sa dodatnom translacijom, rotacijom i refleksijom. To znači da se oba objekta mogu reskalirati, repozicionirati i reflektovati, tako da se tačno poklapaju s drugim objektom. Ako su dva objekta slična, svaki od njih je podudaran sa rezultatom određenog ravnomernog skaliranja drugog.

Na primer, svi krugovi su slični jedni drugima, svi kvadrati su međusobno slični, a svi jednakostranični trouglovi su međusobno slični. S druge strane, elipse nisu slične jedna drugoj, pravougaonici nisu svi međusobno slični, a jednakokraki trouglovi nisu svi slični jedan drugom.

Ako dva ugla trougla imaju mere jednake merama dva ugla drugog trougla, onda su trouglovi slični. Odgovarajuće strane sličnih poligona su proporcionalne, a odgovarajući uglovi sličnih poligona imaju istu meru.

U ovom članku se pretpostavlja da skaliranje može imati faktor skale od 1, tako da su svi podudarni oblici takođe slični, mada neki školski udžbenici eksplicitno isključuju kongruentne trouglove iz njihove definicije sličnih trouglova insistirajući da se veličine moraju razlikovati, ako su trouglovi kvalifikuju kao slični.

Slični trouglovi

U geometriji dva trougla, $△ ABC$ i $△ A'B'C'$ , su slična ako i samo ako korespondirajući uglovi imaju istu meru: to znači da su slični ako i samo ako su dužine odgovarajućih strana proporcionalne.^[1] Može se pokazati da su dva trougla koja imaju kongruentne uglove (jednakougaoni trouglovi) slični, odnosno može se dokazati da su odgovarajuće strane proporcionalne. To je poznato kao AAA teorema sličnosti.^[2] Neophodno je napomenuti da je „AAA” mnemonička konstrukcija: svako od tri „A” odnosi se na jedan ugao. Usled ove teoreme, nekoliko autora pojednostavljuje definiciju sličnih trouglova da bi se samo zahtevalo da su odgovarajuća tri ugla kongruentna.^[3]

Postoji nekoliko tvrdnji od kojih je svaka neophodna i dovoljna da bi dva trougla bila slična:

Trouglovi imaju dva kongruentna ugla,^[4] koji u euklidskoj geometriji podrazumevaju da su svi njihovi uglovi kongruentni.^[5] Drugim rečima:

Ako je

∠ BAC

jednake mere sa

∠ B'A'C'

, i

∠ ABC

jednake mere sa

∠ A'B'C'

, onda se iz toga podrazumeva da je

∠ ACB

jednake mere

∠ A'C'B'

i da su trouglovi slični.

Sve korespondirajuće strane imaju dužine istog odnosa:^[6]

.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num,.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0 0.1em}.mw-parser-output .sfrac .den{border-top:1px solid}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}AB/A′B′ = BC/B′C′ = AC/A′C′

. Ovo je ekvivalentno tvrdnji da je jedan trougao (ili njegov odraz u ogledlu) uvećanje drugog.

Dve strane imaju dužine istog odnosa, i uglovi između ovih strana imaju istu meru.^[7] Na primer:

AB / A'B' = BC / B'C'

i

∠ ABC

ima istu meru sa

∠ A'B'C'

.

Ovo je poznato kao SAS kriterijum sličnosti.^[8] „SAS” je mnemonička konstrukcija: svaki od dva „S” se odnosi na stranicu (engl. side), dok se „A” odnosi na ugao (engl. angle) između stranica.

Kad su dva trougla $△ ABC$ i $△ A'B'C'$ slična, piše se^[9]^{‍:p. 22}

△ ABC \sim △ A'B'C'

.

Postoji nekoliko elementarnih rezultata koji se tiču sličnih trouglova u euklidskoj geometriji:^[10]^[11]

Svaka dva jednakostranična trougla su slična.
Dva trougla, oba slična trećem trouglu, su međusobno slična (tranzitivnost sličnosti trouglova).
Korespondirajuće visine sličnih trouglova imaju isti odnos kao i odgovarajuće strane.
Dva pravougaona trougla su slična ako hipotenuza i jedna druga strana imaju dužine u istom odnosu.^[12]

Polazeći od trougla $△ ABC$ i linijskog segmenta $DE$ , može se pomoću lenjira i šestara pronaći tačka $F$ takva da je $△ ABC \sim △ DEF$ . Tvrdnja da tačka $F$ zadovoljava ovaj uslov postoji kao Valisov postulat^[13] i logički je ekvivalentna Euklidovom postulatu paralelnosti.^[14] U hiperboličkoj geometriji (gde je Valisov postulat ne važi) slični trouglovi su kongurentni.

U aksiomatskom tretmanu euklidske geometrije koji je dao G.D. Birkof (pogledajte Birkofove aksiome),^[15]^[16]^[17] SAS kriterijum sličnosti naveden gore je korišten da se zameni Euklidov postulat paralelnosti i SAS aksiom, što je omogućilo dramatično skraćivanje Hilbertovih aksioma.^[8]

Slični trouglovi pružaju osnovu za mnoge sintetičke (bez upotrebe koordinata) dokaze u euklidskoj geometriji.^[18]^[19] Među elementarnim rezultatima koji se mogu dokazati na ovaj način su: teorema ugaone simetrale,^[20] teorema geometrijske sredine, Čevova teorema, Menelajeva teorema^[21]^[22] i Pitagorina teorema. Slični trouglovi takođe pružaju osvno za trigonometriju pravog trougla.^[23]

U euklidskom prostoru

A similarity (also called a similarity transformation or similitude) of a Euclidean space is a bijection $f$ from the space onto itself that multiplies all distances by the same positive real number $r$ , so that for any two points $x$ and $y$ we have

d(f(x),f(y))=r\,d(x,y),\,

where " $d (x, y)$ " is the Euclidean distance from $x$ to $y$ .^[24] The scalar $r$ has many names in the literature including; the ratio of similarity, the stretching factor and the similarity coefficient. When $r$ = 1 a similarity is called an isometry (rigid transformation). Two sets are called similar if one is the image of the other under a similarity.

As a map $f : ℝ n \to ℝ n$ , a similarity of ratio $r$ takes the form

f(x)=rAx+t,

where $A \in O n (ℝ)$ is an $n \times n$ orthogonal matrix and $t \in ℝ n$ is a translation vector.

Similarities preserve planes, lines, perpendicularity, parallelism, midpoints, inequalities between distances and line segments.^[25] Similarities preserve angles but do not necessarily preserve orientation, direct similitudes preserve orientation and opposite similitudes change it.^[26]

The similarities of Euclidean space form a group under the operation of composition called the similarities group $S$ .^[27] The direct similitudes form a normal subgroup of $S$ and the Euclidean group $E (n)$ of isometries also forms a normal subgroup.^[28] The similarities group $S$ is itself a subgroup of the affine group, so every similarity is an affine transformation.^[29]^[30]

One can view the Euclidean plane as the complex plane,^[31] that is, as a 2-dimensional space over the reals. The 2D similarity transformations can then be expressed in terms of complex arithmetic and are given by $f (z) = az + b$ (direct similitudes) and $f (z) = a z + b$ (opposite similitudes), where $a$ and $b$ are complex numbers, $a \neq 0$ . When $| a | = 1$ , these similarities are isometries.

Reference

^ Sibley 1998, p. 35
^ Stahl 2003, p. 127. This is also proved in Euclid's Elements, Book VI, Proposition 4.
^ For instance, Venema 2006, p. 122 and Henderson & Taimiṇa 2005, p. 123
^ Euclid's elements Book VI Proposition 4.
^ This statement is not true in Non-euclidean geometry where the triangle angle sum is not 180 degrees.
^ Euclid's elements Book VI Proposition 5
^ Euclid's elements Book VI Proposition 6
^ ^а ^б Venema 2006, p. 143
^ Posamentier, Alfred S. and Lehmann, Ingmar. The Secrets of Triangles, Prometheus Books, 2012.
^ Jacobs 1974, pp. 384 - 393
^ Pambuccian, Victor; Schacht, Celia (2021), „The Case for the Irreducibility of Geometry to Algebra”, Philosophia Mathematica, 29 (4), doi:10.1093/philmat/nkab022
^ Hadamard, Jacques (2008), Lessons in Geometry, Vol. I: Plane Geometry, American Mathematical Society, Theorem 120, p. 125, ISBN 9780821843673
^ Named for John Wallis (1616–1703)
^ Venema 2006, p. 122
^ Birkhoff, George David (1932), „A Set of Postulates for Plane Geometry (Based on Scale and Protractors)”, Annals of Mathematics, 33 (2): 329—345, JSTOR 1968336, doi:10.2307/1968336, hdl:10338.dmlcz/147209 
^ Birkhoff, George David; Beatley, Ralph (2000) [first edition, 1940], Basic Geometry (3rd изд.), American Mathematical Society, ISBN 978-0-8218-2101-5
^ Kelly, Paul Joseph; Matthews, Gordon (1981), The non-Euclidean, hyperbolic plane: its structure and consistency, Springer-Verlag, ISBN 0-387-90552-9
^ Ernst Kötter (1901). Die Entwickelung der Synthetischen Geometrie von Monge bis auf Staudt (1847) (PDF). (2012 Reprint as ISBN 1275932649)
^ Pambuccian, Victor; Schacht, Celia (2021), „The Case for the Irreducibility of Geometry to Algebra”, Philosophia Mathematica, 29 (4), doi:10.1093/philmat/nkab022
^ Alfred S. Posamentier: Advanced Euclidean Geometry: Excursions for Students and Teachers. Springer, 2002, ISBN 9781930190856, pp. 3-4
^ Benitez, Julio (2007). „A Unified Proof of Ceva and Menelaus' Theorems Using Projective Geometry” (PDF). Journal for Geometry and Graphics. 11 (1): 39—44.
^ Moussa, Ali (2011). „Mathematical Methods in Abū al-Wafāʾ's Almagest and the Qibla Determinations”. Arabic Sciences and Philosophy. Cambridge University Press. 21 (1): 1—56. S2CID 171015175. doi:10.1017/S095742391000007X.
^ Venema 2006, p. 145
^ Smart 1998, стр. 92.
^ Yale 1968, стр. 47 Theorem 2.1.
^ Pedoe 1988, стр. 179–181.
^ Yale 1968, стр. 46.
^ Pedoe 1988, стр. 182.
^ Nomizu, Katsumi; Sasaki, Takeshi (1994). Affine Differential Geometry: Geometry of Affine Immersions. Cambridge University Press. ISBN 978-0-521-44177-3.
^ Berger, Marcel; Cole, M. (1987). Geometry I. Springer Science & Business Media. ISBN 978-3-540-11658-5.
^ This traditional term, as explained in its article, is a misnomer. This is actually the 1-dimensional complex line.

Literatura

Henderson, David W.; Taimina, Daina (2005), Experiencing Geometry/Euclidean and Non-Euclidean with History (3rd изд.), Pearson Prentice-Hall, ISBN 978-0-13-143748-7
Jacobs, Harold R. (1974), Geometry, W.H. Freeman and Co., ISBN 0-7167-0456-0
Pedoe, Dan (1988) [1970], Geometry/A Comprehensive Course, Dover, ISBN 0-486-65812-0
Sibley, Thomas Q. (1998), The Geometric Viewpoint/A Survey of Geometries, Addison-Wesley, ISBN 978-0-201-87450-1
Smart, James R. (1998), Modern Geometries (5th изд.), Brooks/Cole, ISBN 0-534-35188-3
Stahl, Saul (2003), Geometry/From Euclid to Knots, Prentice-Hall, ISBN 978-0-13-032927-1
Venema, Gerard A. (2006), Foundations of Geometry, Pearson Prentice-Hall, ISBN 978-0-13-143700-5
Yale, Paul B. (1968), Geometry and Symmetry, Holden-Day
Judith N. Cederberg (1989, 2001) A Course in Modern Geometries, Chapter 3.12 Similarity Transformations, pp. 183–9, Springer ISBN 0-387-98972-2. .
H.S.M. Coxeter (1961,9) Introduction to Geometry, §5 Similarity in the Euclidean Plane, pp. 67–76, §7 Isometry and Similarity in Euclidean Space, pp 96–104, John Wiley & Sons.
Günter Ewald (1971) Geometry: An Introduction, pp 106, 181, Wadsworth Publishing.
George E. Martin. Transformation Geometry: An Introduction to Symmetry, Chapter 13 Similarities in the Plane, pp. 136–46, Springer 1982. ISBN 0-387-90636-3..
Boyer, Carl B. (2004) [1956], History of Analytic Geometry, Dover, ISBN 978-0-486-43832-0
Greenberg, Marvin Jay (1974), Euclidean and Non-Euclidean Geometries/Development and History, San Francisco: W.H. Freeman, ISBN 0-7167-0454-4
Halsted, G. B. (1896) Elementary Synthetic Geometry via Internet Archive
Halsted, George Bruce (1906) Synthetic Projective Geometry, via Internet Archive.
Hilbert & Cohn-Vossen, Geometry and the imagination.
Klein, Felix (1948), Elementary Mathematics from an Advanced Standpoint/Geometry, New York: Dover
Mlodinow, Leonard (2001), Euclid's Window/The Story of Geometry from Parallel Lines to Hyperspace, New York: The Free Press, ISBN 0-684-86523-8
Heath, Thomas L. (1956). The Thirteen Books of Euclid's Elements (2nd ed. [Facsimile. Original publication: Cambridge University Press, 1925] изд.). New York: Dover Publications.
Russell, John Wellesley (1905). „Ch. 1 §6 "Menelaus' Theorem"”. Pure Geometry. Clarendon Press.

Spoljašnje veze

Animated demonstration of similar triangles

[1] Sibley 1998, p. 35

[2] Stahl 2003, p. 127. This is also proved in Euclid's Elements, Book VI, Proposition 4.

[3] For instance, Venema 2006, p. 122 and Henderson & Taimiṇa 2005, p. 123

[4] Euclid's elements Book VI Proposition 4.

[5] This statement is not true in Non-euclidean geometry where the triangle angle sum is not 180 degrees.

[6] Euclid's elements Book VI Proposition 5

[7] Euclid's elements Book VI Proposition 6

[Venema_2006_loc=p._143-8] а ^б Venema 2006, p. 143

[PL-9] Posamentier, Alfred S. and Lehmann, Ingmar. The Secrets of Triangles, Prometheus Books, 2012.

[10] Jacobs 1974, pp. 384 - 393

[PS-11] Pambuccian, Victor; Schacht, Celia (2021), „The Case for the Irreducibility of Geometry to Algebra”, Philosophia Mathematica, 29 (4), doi:10.1093/philmat/nkab022

[12] Hadamard, Jacques (2008), Lessons in Geometry, Vol. I: Plane Geometry, American Mathematical Society, Theorem 120, p. 125, ISBN 9780821843673

[13] Named for John Wallis (1616–1703)

[14] Venema 2006, p. 122

[15] Birkhoff, George David (1932), „A Set of Postulates for Plane Geometry (Based on Scale and Protractors)”, Annals of Mathematics, 33 (2): 329—345, JSTOR 1968336, doi:10.2307/1968336, hdl:10338.dmlcz/147209 

[16] Birkhoff, George David; Beatley, Ralph (2000) [first edition, 1940], Basic Geometry (3rd изд.), American Mathematical Society, ISBN 978-0-8218-2101-5

[17] Kelly, Paul Joseph; Matthews, Gordon (1981), The non-Euclidean, hyperbolic plane: its structure and consistency, Springer-Verlag, ISBN 0-387-90552-9

[18] Ernst Kötter (1901). Die Entwickelung der Synthetischen Geometrie von Monge bis auf Staudt (1847) (PDF). (2012 Reprint as ISBN 1275932649)

[19] Pambuccian, Victor; Schacht, Celia (2021), „The Case for the Irreducibility of Geometry to Algebra”, Philosophia Mathematica, 29 (4), doi:10.1093/philmat/nkab022

[20] Alfred S. Posamentier: Advanced Euclidean Geometry: Excursions for Students and Teachers. Springer, 2002, ISBN 9781930190856, pp. 3-4

[21] Benitez, Julio (2007). „A Unified Proof of Ceva and Menelaus' Theorems Using Projective Geometry” (PDF). Journal for Geometry and Graphics. 11 (1): 39—44.

[musa-22] Moussa, Ali (2011). „Mathematical Methods in Abū al-Wafāʾ's Almagest and the Qibla Determinations”. Arabic Sciences and Philosophy. Cambridge University Press. 21 (1): 1—56. S2CID 171015175. doi:10.1017/S095742391000007X.

[23] Venema 2006, p. 145

[FOOTNOTESmart199892-24] Smart 1998, стр. 92.

[FOOTNOTEYale196847_Theorem_2.1-25] Yale 1968, стр. 47 Theorem 2.1.

[FOOTNOTEPedoe1988179–181-26] Pedoe 1988, стр. 179–181.

[FOOTNOTEYale196846-27] Yale 1968, стр. 46.

[FOOTNOTEPedoe1988182-28] Pedoe 1988, стр. 182.

[29] Nomizu, Katsumi; Sasaki, Takeshi (1994). Affine Differential Geometry: Geometry of Affine Immersions. Cambridge University Press. ISBN 978-0-521-44177-3.

[30] Berger, Marcel; Cole, M. (1987). Geometry I. Springer Science & Business Media. ISBN 978-3-540-11658-5.

[31] This traditional term, as explained in its article, is a misnomer. This is actually the 1-dimensional complex line.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[18]

[19]

[20]

[21]

[22]

[23]

[24]

[25]

[26]

[27]

[28]

[29]

[30]

[31]

@@ Ред 27: / Ред 27: @@
 :{{math|△''ABC'' ∼ △''A′B′C′''}}.
-Postoji nekoliko elementarnih rezultata koji se tiču sličnih trouglova u euklidskoj geometriji:<ref>{{harvnb|Jacobs|1974|loc=pp. 384 - 393}}</ref>
+Postoji nekoliko elementarnih rezultata koji se tiču sličnih trouglova u euklidskoj geometriji:<ref>{{harvnb|Jacobs|1974|loc=pp. 384 - 393}}</ref><ref name=PS>{{citation|first1=Victor|last1=Pambuccian|first2=Celia|last2=Schacht|title=The Case for the Irreducibility of Geometry to Algebra|journal=Philosophia Mathematica|volume=29|year=2021|issue=4|doi=10.1093/philmat/nkab022|url=https://academic.oup.com/philmat/advance-article-abstract/doi/10.1093/philmat/nkab022/6371269?redirectedFrom=fulltext}}</ref>
 * Svaka dva [[Једнакостранични троугао|jednakostranična trougla]] su slična.
 * Dva trougla, oba slična trećem trouglu, su međusobno slična ([[Transitive relation|tranzitivnost]] sličnosti trouglova).
@@ Ред 35: / Ред 35: @@
 Polazeći od trougla {{math|△''ABC''}} i linijskog segmenta {{math|{{overline|''DE''}}}}, može se pomoću [[Конструкције лењиром и шестаром|lenjira i šestara]] pronaći tačka {{math|''F''}} takva da je {{math|△''ABC'' ∼ △''DEF''}}. Tvrdnja da tačka {{math|''F''}} zadovoljava ovaj uslov postoji kao Valisov postulat<ref>Named for [[John Wallis]] (1616–1703)</ref> i logički je ekvivalentna [[Постулат паралелности|Euklidovom postulatu paralelnosti]].<ref>{{harvnb|Venema|2006|loc=p. 122}}</ref> U [[Хиперболичка геометрија|hiperboličkoj geometriji]] (gde je Valisov postulat ne važi) slični trouglovi su kongurentni.
-U aksiomatskom tretmanu euklidske geometrije koji je dao [[George David Birkhoff|G.D. Birkof]] (pogledajte [[Birkhoff's axioms|Birkofove aksiome]]), SAS kriterijum sličnosti naveden gore je korišten da se zameni Euklidov postulat paralelnosti i SAS aksiom, što je omogućilo dramatično skraćivanje [[Хилбертове аксиоме|Hilbertovih aksioma]].<ref name="Venema 2006 loc=p. 143"/>
+U aksiomatskom tretmanu euklidske geometrije koji je dao [[George David Birkhoff|G.D. Birkof]] (pogledajte [[Birkhoff's axioms|Birkofove aksiome]]),<ref>{{citation|last=Birkhoff|first= George David|author-link=George David Birkhoff|year= 1932|title=A Set of Postulates for Plane Geometry (Based on Scale and Protractors)|journal= [[Annals of Mathematics]]|volume= 33|issue= 2|pages=329–345|doi=10.2307/1968336|jstor= 1968336|hdl=10338.dmlcz/147209|hdl-access=free}}</ref><ref>{{citation|last1=Birkhoff|first1= George David|first2=Ralph|last2= Beatley|author1-link=George David Birkhoff|title=Basic Geometry|edition= 3rd|publisher=American Mathematical Society|year=2000|isbn=978-0-8218-2101-5|orig-year=first edition, 1940}}</ref><ref>{{citation|first1=Paul Joseph|last1= Kelly|first2=Gordon|last2=Matthews|year= 1981|title=The non-Euclidean, hyperbolic plane: its structure and consistency|isbn=0-387-90552-9|publisher=Springer-Verlag}}</ref> SAS kriterijum sličnosti naveden gore je korišten da se zameni Euklidov postulat paralelnosti i SAS aksiom, što je omogućilo dramatično skraćivanje [[Хилбертове аксиоме|Hilbertovih aksioma]].<ref name="Venema 2006 loc=p. 143"/>
-Slični trouglovi pružaju osnovu za mnoge [[Synthetic geometry|sintetičke]] (bez upotrebe koordinata) dokaze u euklidskoj geometriji. Među elementarnim rezultatima koji se mogu dokazati na ovaj način su: [[Angle bisector theorem|teorema ugaone simetrale]], [[geometric mean theorem|teorema geometrijske sredine]], [[Ceva's theorem|Čevova teorema]], [[Menelaus's theorem|Menelajeva teorema]] i [[Pitagorina teorema]]. Slični trouglovi takođe pružaju osvno za [[Trigonometrija|trigonometriju pravog trougla]].<ref>{{harvnb|Venema|2006|loc=p. 145}}</ref>
+Slični trouglovi pružaju osnovu za mnoge [[Synthetic geometry|sintetičke]] (bez upotrebe koordinata) dokaze u euklidskoj geometriji.<ref>{{cite book| author=Ernst Kötter| title=Die Entwickelung der Synthetischen Geometrie von Monge bis auf Staudt (1847)| year=1901 |url=http://gdz-lucene.tc.sub.uni-goettingen.de/gcs/gcs?&action=pdf&metsFile=PPN37721857X_0005&divID=LOG_0035&pagesize=original&pdfTitlePage=http://gdz.sub.uni-goettingen.de/dms/load/pdftitle/?metsFile=PPN37721857X_0005%7C&targetFileName=PPN37721857X_0005_LOG_0035.pdf&}} (2012 Reprint as {{ISBN|1275932649}})</ref><ref>{{Citation|first1=Victor|last1=Pambuccian|first2=Celia|last2=Schacht|title=The Case for the Irreducibility of Geometry to Algebra|journal=Philosophia Mathematica|volume=29|year=2021 |issue=4|doi=10.1093/philmat/nkab022|url=https://academic.oup.com/philmat/advance-article-abstract/doi/10.1093/philmat/nkab022/6371269?redirectedFrom=fulltext}}</ref> Među elementarnim rezultatima koji se mogu dokazati na ovaj način su: [[Angle bisector theorem|teorema ugaone simetrale]],<ref>Alfred S. Posamentier: ''Advanced Euclidean Geometry: Excursions for Students and Teachers''. Springer, 2002, {{ISBN|9781930190856}}, pp. [https://books.google.com/books?id=9grsxFZUci8C&pg=PA4 3-4]</ref> [[geometric mean theorem|teorema geometrijske sredine]], [[Ceva's theorem|Čevova teorema]], [[Menelaus's theorem|Menelajeva teorema]]<ref>{{Cite journal |last=Benitez |first=Julio |date=2007 |title=A Unified Proof of Ceva and Menelaus' Theorems Using Projective Geometry |url=https://www.heldermann-verlag.de/jgg/jgg11/j11h1beni.pdf |journal=Journal for Geometry and Graphics |volume=11 |issue=1 |pages=39–44}}</ref><ref name="musa">{{cite journal|last=Moussa|first=Ali|title=Mathematical Methods in Abū al-Wafāʾ's Almagest and the Qibla Determinations|journal=Arabic Sciences and Philosophy|year=2011|volume=21|issue=1|pages=1–56|publisher=[[Cambridge University Press]]|doi=10.1017/S095742391000007X|s2cid=171015175}}</ref> i [[Pitagorina teorema]]. Slični trouglovi takođe pružaju osvno za [[Trigonometrija|trigonometriju pravog trougla]].<ref>{{harvnb|Venema|2006|loc=p. 145}}</ref>
+== U euklidskom prostoru ==
+{{rut}}
+A '''similarity''' (also called a '''similarity transformation''' or '''similitude''') of a [[Euclidean space]] is a [[bijection]] {{math|''f''}} from the space onto itself that multiplies all distances by the same positive [[real number]] {{math|''r''}}, so that for any two points {{math|''x''}} and {{math|''y''}} we have
+:<math>d(f(x),f(y)) = r\, d(x,y), \, </math>
+where "{{math|''d''(''x'',''y'')}}" is the [[Euclidean distance]] from {{math|''x''}} to {{math|''y''}}.{{sfn|Smart|1998|p=92}} The [[scalar (mathematics)|scalar]] {{math|''r''}} has many names in the literature including; the ''ratio of similarity'', the ''stretching factor'' and the ''similarity coefficient''. When {{math|''r''}} = 1 a similarity is called an [[Euclidean plane isometry|isometry]] ([[rigid transformation]]). Two sets are called '''similar''' if one is the image of the other under a similarity.
+As a map {{math|''f'' : ℝ{{sup|''n''}} → ℝ{{sup|''n''}}}}, a similarity of ratio {{math|''r''}} takes the form
+:<math>f(x) = rAx + t,</math>
+where {{math|''A'' ∈ ''O''{{sub|''n''}}(ℝ)}} is an {{math|''n'' × ''n''}} [[orthogonal matrix]] and {{math|''t'' ∈ ℝ{{sup|''n''}}}} is a translation vector.
+Similarities preserve planes, lines, perpendicularity, parallelism, midpoints, inequalities between distances and line segments.{{sfn|Yale|1968|p=47 Theorem 2.1}} Similarities preserve angles but do not necessarily preserve orientation, ''direct similitudes'' preserve orientation and ''opposite similitudes'' change it.{{sfn|Pedoe|1988|pp=179–181}}
+The similarities of Euclidean space form a [[group (mathematics)|group]] under the operation of composition called the ''similarities group ''{{math|''S''}}.{{sfn|Yale|1968|p=46}} The direct similitudes form a [[normal subgroup]] of {{math|''S''}} and the [[Euclidean group]] {{math|''E''(''n'')}} of isometries also forms a normal subgroup.{{sfn|Pedoe|1988|p=182}} The similarities group {{math|''S''}} is itself a subgroup of the [[affine group]], so every similarity is an [[affine transformation]].<ref>{{Cite book| ref=harv|last=Nomizu|first=Katsumi|last2=Sasaki|first2=Takeshi|title=Affine Differential Geometry: Geometry of Affine Immersions|url=https://books.google.com/books?id=lEUVHyjQANcC|year=1994|publisher=Cambridge University Press|isbn=978-0-521-44177-3|pages=}}</ref><ref>{{Cite book| ref=harv|last=Berger |first=Marcel|last2=Cole|first2=M.|title=Geometry I|url=https://books.google.com/books?id=5W6cnfQegYcC|year=1987|publisher=Springer Science & Business Media|isbn=978-3-540-11658-5|pages=}}</ref>
+One can view the Euclidean plane as the [[complex plane]],<ref>This traditional term, as explained in its article, is a misnomer. This is actually the 1-dimensional complex line.</ref> that is, as a 2-dimensional space over the [[real number|reals]]. The 2D similarity transformations can then be expressed in terms of complex arithmetic and are given by {{math|''f''(''z'') {{=}} ''az'' + ''b''}} (direct similitudes) and {{math|''f''(''z'') {{=}} ''a{{overline|z}}'' + ''b''}} (opposite similitudes), where {{math|''a''}} and {{math|''b''}} are complex numbers, {{math|''a'' ≠ 0}}. When {{math|{{!}}''a''{{!}} {{=}} 1}}, these similarities are isometries.
 == Reference ==
@@ Ред 56: / Ред 76: @@
 * Günter Ewald (1971) ''Geometry: An Introduction'', pp 106, 181, [[Wadsworth Publishing]].
 * George E. Martin. ''Transformation Geometry: An Introduction to Symmetry'', Chapter 13 Similarities in the Plane, pp. 136–46, Springer {{page|year=1982|isbn=0-387-90636-3|pages=}}.
+* {{citation|first=Carl B.|last=Boyer|title=History of Analytic Geometry|year=2004|orig-year=1956|publisher=Dover|isbn=978-0-486-43832-0}}
+* {{citation|author-link=Marvin Greenberg|first=Marvin Jay|last=Greenberg|title=Euclidean and Non-Euclidean Geometries/Development and History|year=1974|publisher=W.H. Freeman|place=San Francisco|isbn=0-7167-0454-4}}
+* Halsted, G. B. (1896) [https://archive.org/details/elementarysynth00halsgoog Elementary Synthetic Geometry] via Internet Archive
+* [[G. B. Halsted|Halsted, George Bruce]] (1906) [https://archive.org/details/syntheticproject00halsuoft Synthetic Projective Geometry], via [[Internet Archive]].
+* Hilbert & Cohn-Vossen, ''Geometry and the imagination''.
+* {{citation|first=Felix|last=Klein|title=Elementary Mathematics from an Advanced Standpoint/Geometry|publisher=Dover|place=New York|year=1948}}
+* {{citation|first=Leonard|last=Mlodinow|title=Euclid's Window/The Story of Geometry from Parallel Lines to Hyperspace|publisher=The Free Press|place=New York|year=2001|isbn=0-684-86523-8|url-access=registration|url=https://archive.org/details/euclidswindowsto00mlod}}
+* {{cite book | last = Heath | first = Thomas L. | author-link = T. L. Heath | title = The Thirteen Books of Euclid's Elements | url = https://archive.org/details/thirteenbooksofe00eucl | url-access = registration | edition = 2nd ed. [Facsimile. Original publication: Cambridge University Press, 1925] | year = 1956 | publisher = Dover Publications | location = New York }}
+* {{cite book |title=Pure Geometry |first=John Wellesley|last=Russell|publisher=Clarendon Press |year=1905 |chapter= Ch. 1 §6 "Menelaus' Theorem" |url=https://archive.org/details/anelementarytre02russgoog}}
 {{refend}}