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[{"key": "dc.contributor.author", "value": "Suomela, Maria", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2020-02-03T06:30:35Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2020-02-03T06:30:35Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2020", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/67690", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "Tutkielman tarkoituksena on perehdytt\u00e4\u00e4 lukija vektoreiden pohjalta luotuun geometriaan. Monesti geometriasta puhuttaessa tulee ensimm\u00e4isen\u00e4 mieleen aksiomaattinen geometria kuten Eukleideen tai Hilbertin luomat aksiomaattiset j\u00e4rjestelm\u00e4t, mutta t\u00e4ss\u00e4 tutkielmassa n\u00e4k\u00f6kulma on erilainen. Geometriaa on alettu kehitt\u00e4m\u00e4\u00e4n jo ennen ajanlaskun alkua, mutta vektoreihin perustuva geometria on kehitetty vasta 1800-luvun loppupuolella. \n\nTutkielma jakautuu kolmeen osaan, jossa ensimm\u00e4inen osa k\u00e4sittelee yhdensuuntaisia suoria ja eri suorilla olevien pisteiden v\u00e4lisi\u00e4 jakosuhteita. Suorat ovat geometriassa usein l\u00e4ht\u00f6kohtana ja siksi niihin on hyv\u00e4 syventy\u00e4 ennen geometrian muita osa-alueita. Monissa t\u00e4m\u00e4n tutkielman todistuksissa otetaan apuv\u00e4lineeksi k\u00e4ytt\u00f6\u00f6n jakosuhteet, joten aluksi on hyv\u00e4 tutustua niiden ominaisuuksiin. \n\nToisessa osassa keskityt\u00e4\u00e4n kolmioihin ja niiden ominaisuuksiin. Kolmen eri suoran leikkauspisteist\u00e4 saadaan aikaan kolmio, joten suoriin perehtymisen j\u00e4lkeen kolmio on jatkumoa edelliselle osalle. Kolmioille on olemassa monia tunnettuja lauseita, joita t\u00e4ss\u00e4kin tutkielmassa tarkastellaan. N\u00e4it\u00e4 ovat muun muassa Cevan lause sek\u00e4 kolmioiden yhtenevyys- ja yhdenmuotoisuuss\u00e4\u00e4nn\u00f6t.\n\nKolmas osa jatkaa kolmioiden ominaisuuksista, sill\u00e4 osassa tutustutaan ja perehdyt\u00e4\u00e4n kolmion merkillisiin pisteisiin. Kolmion merkilliset pisteet syntyv\u00e4t kolmion sivujen ja/tai kulmien v\u00e4lille muodostuvien janojen tai niiden jatkeiden leikkauspisteist\u00e4. T\u00e4ss\u00e4 tutkielmassa esitell\u00e4\u00e4n nelj\u00e4 kolmion merkillist\u00e4 pistett\u00e4, joista ensimm\u00e4inen muodostuu kolmion korkeusjanojen leikkauspisteen\u00e4, toinen on kolmion keskijanojen leikkauspiste ja kolmas kolmion merkillinen piste syntyy kolmion keskinormaalien leikkauspisteist\u00e4. N\u00e4iden kolmen kolmion merkillisen pisteen kautta kulkee Eulerin suoraksi kutsuttu suora. Nelj\u00e4s kolmion merkillisist\u00e4 pisteist\u00e4, joka ei sis\u00e4lly Eulerin suoraan, muodostuu kolmion kulmanpuolittajien leikkauspisteeseen.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Miia Hakanen (mihakane@jyu.fi) on 2020-02-03T06:30:35Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2020-02-03T06:30:35Z (GMT). No. of bitstreams: 0\n Previous issue date: 2020", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "40", "language": "", "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.format.mimetype", "value": "application/pdf", "language": null, "element": "format", "qualifier": "mimetype", "schema": "dc"}, {"key": "dc.language.iso", "value": "fin", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": "en", "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.subject.other", "value": "Cevan lause", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "yhtenevyy", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "yhdemuotoisuus", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "kolmion merkilliset pisteet", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Eulerin suora", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Geometriaa vektoreilla", "language": "", "element": "title", "qualifier": null, "schema": "dc"}, {"key": "dc.type", "value": "master thesis", "language": null, "element": "type", "qualifier": null, "schema": "dc"}, {"key": "dc.identifier.urn", "value": "URN:NBN:fi:jyu-202002031954", "language": "", "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu -tutkielma", "language": "fi", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Master\u2019s thesis", "language": "en", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Matemaattis-luonnontieteellinen tiedekunta", "language": "fi", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Faculty of Sciences", "language": "en", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Matematiikan ja tilastotieteen laitos", "language": "fi", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Department of Mathematics and Statistics", "language": "en", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "Jyv\u00e4skyl\u00e4n yliopisto", "language": "fi", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": "en", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Matematiikan opettajankoulutus", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Teacher education programme in Mathematics", "language": "en", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "yvv.contractresearch.funding", "value": "0", "language": "", "element": "contractresearch", "qualifier": "funding", "schema": "yvv"}, {"key": "dc.type.coar", "value": "http://purl.org/coar/resource_type/c_bdcc", "language": null, "element": "type", "qualifier": "coar", "schema": "dc"}, {"key": "dc.rights.accesslevel", "value": "openAccess", "language": null, "element": "rights", "qualifier": "accesslevel", "schema": "dc"}, {"key": "dc.type.publication", "value": "masterThesis", "language": null, "element": "type", "qualifier": "publication", "schema": "dc"}, {"key": "dc.subject.oppiainekoodi", "value": "4041", "language": "", "element": "subject", "qualifier": "oppiainekoodi", "schema": "dc"}, {"key": "dc.subject.yso", "value": "vektorit (matematiikka)", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "matematiikka", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "geometria", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.format.content", "value": "fulltext", "language": null, "element": "format", "qualifier": "content", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}, {"key": "dc.type.okm", "value": "G2", "language": null, "element": "type", "qualifier": "okm", "schema": "dc"}]
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