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src=\"//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Treffpunkt.svg/26px-Treffpunkt.svg.png\" decoding=\"async\" width=\"26\" height=\"26\" class=\"mw-file-element\" srcset=\"//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Treffpunkt.svg/39px-Treffpunkt.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/97/Treffpunkt.svg/52px-Treffpunkt.svg.png 2x\" data-file-width=\"512\" data-file-height=\"512\" /\u003E\u003C/span\u003E\u003C/span\u003E\u003C/td\u003E\u003Ctd class=\"mbox-text\"\u003EOtvoren je \u003Ca href=\"/wiki/Wikipedija:Zahtjev_za_komentare/Nazivi_%C4%8Dlanaka_o_pojedinim_razdobljima_Hrvatskoga_Kraljevstva\" title=\"Wikipedija:Zahtjev za komentare/Nazivi članaka o pojedinim razdobljima Hrvatskoga Kraljevstva\"\u003E\u003Cb\u003EZahtjev za komentare o nazivima članaka o razdobljima Hrvatskoga Kraljevstva\u003C/b\u003E\u003C/a\u003E\u003C/td\u003E\u003C/tr\u003E\u003C/tbody\u003E\u003C/table\u003E\n\u003Clink rel=\"mw-deduplicated-inline-style\" 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članaka\u003C/b\u003E\u003C/a\u003E\u003C/td\u003E\u003C/tr\u003E\u003C/tbody\u003E\u003C/table\u003E\u003C/div\u003E\u003C/div\u003E\u003C/div\u003E\u003C/div\u003E";}}());</script></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Mrežno mjesto"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <h1 id="firstHeading" class="firstHeading mw-first-heading">Vidi kôd stranice Trokut</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Članak je napisan samo na 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</div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Pomagala"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Izgled"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Izgled</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">premjesti</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">sakrij</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> </div> <div id="contentSub"><div id="mw-content-subtitle">← <a href="/wiki/Trokut" title="Trokut">Trokut</a></div></div> <div id="mw-content-text" class="mw-body-content"><p>Nemate dopuštenje za uređivanje ove stranice iz navedenog razloga: </p> <div class="permissions-errors"><div class="mw-permissionerror-globalblocking-blockedtext-range"><b>Your IP address is in a range that has been <a href="https://meta.wikimedia.org/wiki/Special:MyLanguage/Global_blocks" class="extiw" title="m:Special:MyLanguage/Global blocks">blocked on all Wikimedia Foundation wikis</a>.</b> <p>The block was made by <a 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The reason given is <i><a href="https://meta.wikimedia.org/wiki/Special:MyLanguage/NOP" class="extiw" title="m:Special:MyLanguage/NOP">Open proxy/Webhost</a>: See the <a href="https://meta.wikimedia.org/wiki/WM:OP/H" class="extiw" title="m:WM:OP/H">help page</a> if you are affected </i>. </p> <ul><li>Start of block: 16:12, 27. kolovoza 2023.</li> <li>Expiry of block: 16:12, 27. kolovoza 2028.</li></ul> <p>Your current IP address is 8.222.208.146. The blocked range is 8.222.128.0/17. </p><p>Please include all above details in any queries you make. </p> If you believe you were blocked by mistake, you can find additional information and instructions in the <a href="https://meta.wikimedia.org/wiki/Special:MyLanguage/Stewards/Wizard" class="extiw" title="m:Special:MyLanguage/Stewards/Wizard">Stewards Block Wizard</a>.</div></div><hr /> <p>Možete pogledati i kopirati izvorni sadržaj ove stranice. </p><textarea readonly="" accesskey="," id="wpTextbox1" cols="80" rows="25" style="" class="mw-editfont-monospace" lang="hr" dir="ltr" name="wpTextbox1">{{dz}} [[Datoteka:Triangle illustration.svg|mini|Trokut]] '''Trokut''' je [[geometrijski lik]] koji ima 3 [[stranica (geometrija)|stranice]], 3 [[kut]]a i 3 [[vrh (geometrija)|vrha]]. Oznaka za trokut s vrhovima u točkama A, B i C je <math>\triangle</math>ABC. ==Vrste== Trokuti se prema vrsti kutova dijele na pravokutne, šiljastokutne i tupokutne: * '''Pravokutan''' trokut ima jedan [[pravi kut]] * '''Šiljastokutan''' trokut ima sve kutove šiljaste * '''Tupokutan''' trokut ima jedan tupi kut. Trokuti se nazivaju i prema odnosu duljina stranica; za neke se kaže da su jednakostranični, za neke da su jednakokračni, a svi ostali su raznostranični''':''' * '''[[Jednakostranični trokut|Jednakostranični]]''' trokut je onaj kome su sve stranice istih dužina * '''Jednakokračni''' trokut je onaj kome su dvije stranice istih duǉina, i te stranice se nazivaju krakovi, dok je treća stranica (osnovica) različite duǉine od duǉine krakova * '''Raznostraničan''' trokut je onaj trokut kome su sve stranice različitih duljina. <gallery widths="120"> Datoteka:Triangle.Equilateral.svg| Jednakostraničan trokut Datoteka:Triangle.Isosceles.svg| Jednakokračan, šiljastokutan trokut Datoteka:Triangle.Scalene.svg| Raznostraničan, tupokutan trokut </gallery> == Svojstva == [[Datoteka:Triangle with notations (black).svg|mini|Uobičajene oznake vrhova trokuta (ABC), pripadnih kutova (''αβγ'') i njima nasuprotnih stranica (''abc'')]] [[Datoteka:Triangle.Orthocenter.png|mini|Visine trokuta]] [[Datoteka:Triangle.Centroid.svg|mini|Težišnice trokuta]] Vrhovi trokuta su točke A, B i C. Njima nasuprotne stranice su dužine BC, CA i AB duljina a, b i c. Te stranice u vrhovima A, B i C zatvaraju kutove ''α, β i γ.'' Visina trokuta iz danog vrha je dužina okomita na pravac na kojem leži nasuprotna stranica čiji je drugi kraj na tom pravcu; u tupokutnom trokutu sve visine ne završavaju na stranici trokuta nego na njenom produljenju. Težišnica trokuta je dužina koja spaja vrh i polovište nasuprotne stranice. Srednjica trokuta je [[dužina]] koja spaja polovišta dviju njegovih stranica. ===Kutovi=== Svojstvo [[kut]]ova trokuta je da se nasuprot većoj stranici nalazi veći kut, a nasuprot manjoj stranici se nalazi manji kut. Zahvaljujući tom svojstvu možemo zaključiti puno o trokutima: kod jednakostraničnog trokuta imamo i sve jednake kutove 60°, kod jednakokračnog trokuta imamo 2 jednaka i 1 različit kut, a kod raznostraničnog trokuta imamo tri različita kuta. Zbroj sva tri unutarnja kuta u trokutu uvijek iznosi '''α + β + γ = 180°'''. Zbroj sva tri vanjska kuta u trokutu uvijek iznosi '''α<sub>1</sub> + β<sub>1</sub> + γ<sub>1</sub> = 360°'''. Prema pravilu zbroja kutova u trokutu vrijedi ''α<sub>1</sub> = β + γ'', ''β<sub>1</sub> = α + γ'', te ''γ<sub>1</sub> = α + β''. === Stranice === U trokutu je zbroj duljina bilo kojih dviju stranica uvijek veći od duljine treće stranice:<math display="block">\begin{array}{lcl} a+b &\ge& c \\ b+c &\ge& a \\ c+a &\ge& b \end{array}</math>U pravokutnom trokutu stranice uz pravi kut zovu se katete, a stranica nasuprot pravom kutu je hipotenuza. Za njihove duljine vrijedi [[Pitagorin poučak]] – zbroj kvadrata duljina kateta (a i b) jednak je kvadratu duljine hipotenuze (c):<math display="block">a^2+b^2=c^2</math>I općenito, za duljine stranica svih trokuta vrijedi [[Kosinusni poučak|poučak o kosinusima]], ovdje iskazan samo za kut nasuprot stranice c: <math display="block">c^2 = a^2+b^2-2\,a\,b \cos\gamma</math> === Opseg === [[Opseg]] trokuta je zbroj duǉina svih njegovih stranica: * za jednakostranični trokut opseg je '''3a''', gdje je ''a'' duljina stranice * za jednakokračni trokut opseg je '''2a + b''', gdje je ''a'' duljina kraka, a ''b'' duǉina osnovice * za raznostranični trokut opseg je '''a + b + c''', gdje su ''a'', ''b'' i ''c'' duǉine pojedinih stranica. ===Površina=== Površina trokuta ''P'' računa se kao: :<math>P=\frac{1}{2}a v_a</math> gdje je ''a'' duljina stranice, a ''v<sub>a</sub>'' duljina visine nad tom stranicom.<br>Površinu možemo računati i po Heronovoj formuli (Heronov obrazac), uz uporabu kratice za poluopseg trokuta <math>s=\frac{1}{2} (a+b+c)</math>: :<math>P=\sqrt{s \cdot (s-a) \cdot (s-b) \cdot (s-c)}</math> :<math>P=\frac{1}{4}\sqrt{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}</math> :<math>P=\frac{1}{4}\sqrt{(a^2+b^2+c^2)^2-2(a^4+b^4+c^4)}</math> :<math>P=\frac{1}{4}\sqrt{4a^2b^2-(a^2+b^2-c^2)^2}.</math> Za površinu trokuta vrijede i formule: :<math>P = s \cdot r</math> :<math>P = \frac{abc}{4R}</math> gdje je ''R'' duljina polumjera trokutu opisane kružnice, a ''r'' duljina polumjera trokutu upisane kružnice. ====Površina pravokutnog trokuta==== [[Datoteka:Right triangle with notations (black).svg|mini|Pravokutni trokut]] Površina pravokutnog trokuta se računa prema formuli: <math>P=\frac{1}{2} a\cdot b</math> U ovoj formuli, ''a'' i ''b'' su ''katete'' pravokutnog trokuta, odnosno stranice trokuta koje zatvaraju pravi kut. Formula se izvodi iz činjenice da stranice pravokutnika s jednom dijagonalom pravokutnika čine pravokutan trokut, gdje su stranice pravokutnika katete pravokutnog trokuta, a dijagonala pravokutnika ''hipotenuza'' pravokutnog trokuta. Na taj način od jednog pravokutnika dobivamo dva sukladna pravokutna trokuta, pa je i površina dobivenog trokuta dva puta manja od površine pravokutnika, koja iznosi ''ab''. === Karakteristične točke i pravci trokuta === [[File:Triangle.EulerLine.svg|thumb|right|297px|Težišnice trokuta (narančasto), ortocentar (plavo), središte trokutu opisane kružnice (zeleno) te Eulerov pravac (crveno)]] Dva neparalelna pravca sijeku se u jednoj točki. Kad promatramo tri pravca onda se svaka dva sijeku u po jednoj točki, a vrlo je rijetko će te tri točke biti zapravo jedna te ista. Ako promatramo tri pravca na kojima leže visine, ili tri težišnice, ili tri simetrale kuta, ili tri simetrale stranica trokuta, može se pokazati da svaka od te četiri trojke pravaca/dužina ima jedno sjecište svih triju pravaca. Ta sjecišta su 4 osobite (karakteristične) točke trokuta. Ako je trokut istostraničan te se 4 točke i međusobno podudaraju, a inače se sve 4 međusobno razlikuju. ''Težišnica trokuta'' spaja vrh s polovištem nasuprotne stranice. Tri težišnice sijeku se u točki koja se naziva '''težište trokuta'''. Težište trokuta dijeli svaku težišnicu u [[omjer]]u 2:1, mjereći od vrha trokuta.<ref name="hren" /> Težište trokuta u omjeru 2:1 dijeli i dužinu koja spaja ortocentar i središte trokutu opisane kružnice, mjereći od ortocentra. Po teoremu o ortocentru, pravci na kojima leže tri visine trokuta sijeku se u jednoj točki (neki autori visinama zovu te pravce; kod tupokutnog trokuta ta točka nije na visinama shvaćenim kao dužinama nego na njihovim produžecima). Tu točku presjeka zovemo '''[[ortocentar]]'''. Postoji jedinstvena kružnica koja (iznutra) dira sve tri stranice trokuta, trokutu upisana kružnica. '''Središte trokutu upisane kružnice''' sjecište je simetrala unutarnjih [[kut]]ova trokuta. Simetrale dvaju vanjskih kutova pri dvama vrhovima i simetrala unutarnjega kuta pri trećem vrhu trokuta sijeku se u ''središtu trokutu pripisane kružnice'', tj. središtu kružnice koja izvana dira jednu stranicu trokuta i produljenja po pravcu drugih dviju stranica. Dakle trokutu su pripisane tri pripisane kružnice.<ref name="hren" /> Postoji jedinstvena kružnica koja prolazi kroz sva tri vrha trokuta, trokutu opisana kružnica. '''Središte trokutu opisane kružnice''' je sjecište [[simetrala dužine|simetrala stranica]] trokuta. Ako je trokut šiljastokutan, to središte se nalazi unutar trokuta, ako je trokut pravokutan, na [[polovište|polovištu]] hipotenuze, a ako je trokut tupokutan, nalazi se izvan trokuta. '''Eulerov pravac''' je [[pravac]] na kojemu leže tri središta trokuta: težište, središte opisane kružnice i ortocentar. Jednoznačno je određen za svaki trokut osim jednakostraničnoga.<ref name="hren">{{citiranje weba |url=http://www.enciklopedija.hr/natuknica.aspx?ID=62433 |work=Hrvatska enciklopedija |title=Trokut |publisher=Leksikografski zavod Miroslav Krleža |accessdate=31. srpnja 2016.}}</ref> Središte trokutu upisane kružnice nikad ne leži na Eulerovom pravcu, osim ako je trokut jednakokračan ili jednakostraničan.<ref>{{citiranje weba |url=https://www.youtube.com/watch?v=wVH4MS6v23U |title=Triangles have a Magic Highway - Numberphile |language=engleski |publisher=[[YouTube]]}}</ref> '''[[Feuerbachova kružnica]]''' (kružnica 9 točaka) prolazi kroz nožišta visina, polovišta stranica i polovišta dužina vrh-ortocentar. '''Symmedijana''' je pravac osnosimetričan na težišnicu s obzirom na simetralu odgovarajućeg kuta. Sve tri symmedijane sijeku se u Lemoinovoj točki. '''Simsonov pravac''' je pravac na kojem se nalaze nožišta okomica spuštenih na stranice trokuta (ili njihova produljenja) iz bilo koje točke trokutu opisane kružnice. == Sukladnost == Dva su podskupa ravnine [[sukladnost (geometrija)|sukladna]] ako postoji izometrija koja bijektivno preslikava jedan podskup na drugi. Karakterizacija (u nekim pristupima definicija) sukladnosti za slučaj dva trokuta je da su dva trokuta sukladna ako su im jednake odgovarajuće stranice, <math>a=a', b=b', c=c'</math>, i istovremeno odgovarajući kutovi, <math>\alpha = \alpha',\beta = \beta', \gamma=\gamma'</math>. Osnovnom teorem o izometrijama ravnine govori da je svaka izometrija ravnine kompozicija najviše tri osne simetrije. Dakle, bilo koja dva sukladna trokuta se mogu preslikati jedan od drugog kompozicijom najviše tri [[osna simetrija|osne simetrije]]. Alternativno, možemo jedan od vrhova premjestiti [[translacija|translacijom]] u vrh drugog trokuta koji mu po sukladnosti odgovara, namjestiti potom jednu od stranica iz tog vrha [[vrtnja|rotacijom]] oko tog drugog vrha i još jednom osnom simetrijom okrenuti trokut oko te stranice ako se trokut još uvijek razlikuje od traženog. Naime dva sukladna trokuta sa zajedničkom stranicom su ili isti skup točaka ili se jedan od drugog dobije osnom simetrijom u odnosu na tu stranicu. Sukladnost dvaju trokuta se često dokazuje poučcima o sukladnosti trokuta: * '''S-S-S''' (stranica-stranica-stranica) Dva su trokuta sukladna ako su za ta dva trokuta jednake sve tri stranice. * '''K-S-K''' (kut-stranica-kut) Dva su trokuta sukladna ako im je jednaka jedna stranica i njoj priležeći kutovi. * '''S-K-S''' (stranica-kut-stranica) Dva su trokuta sukladna ako su im jednake dvije stranice i kut između njih. * '''S-S-K''' (stranica-stranica-kut) Dva su trokuta sukladna ako su im jednake dvije stranice i kut nasuprot većoj stranici. == Sličnost == Dva ili više trokuta mogu biti '''slični'''. Slični trokuti imaju sva tri kuta jednaka. Posljedično, sve im se stranice odnose u istom omjeru: <math>a_1 : a'_1 = a_2 : a'_2 = a_3 : a'_3 = k</math> Površine im se odnose kao: <math>P : P ' = k^2</math> Ako su dva trokuta slična, tada se iz jednoga translacijom, rotacijom, refleksijom i [[skaliranje]]m može dobiti drugi. === Poučci o sličnosti trokuta === Sličnost se dokazuje poučcima o sličnosti: * S-S-S, tj. stranica — stranica — stranica. Prema tom poučku, trokuti su slični ako im se sve odgovarajuće stranice odnose u istom omjeru. * S-K-S, tj. stranica — kut — stranica. Prema tom poučku, trokuti su slični ako im se jedan par odgovarajućih stranica odnosi u istom omjeru, te im je kut između tih stranica jednak. * K-K, tj. kut — kut. 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id="ca-viewsource-sticky-header" tabindex="-1" data-event-name="ve-edit-protected-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-editLock mw-ui-icon-wikimedia-wikimedia-editLock"></span> <span></span> </a> </div> <div class="vector-sticky-header-buttons"> <button class="cdx-button cdx-button--weight-quiet mw-interlanguage-selector" id="p-lang-btn-sticky-header" tabindex="-1" data-event-name="ui.dropdown-p-lang-btn-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-language mw-ui-icon-wikimedia-wikimedia-language"></span> <span>Dodajte druge jezike</span> </button> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive" id="ca-addsection-sticky-header" tabindex="-1" data-event-name="addsection-sticky-header"><span class="vector-icon mw-ui-icon-speechBubbleAdd-progressive mw-ui-icon-wikimedia-speechBubbleAdd-progressive"></span> <span>Dodaj temu</span> </a> </div> <div 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