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[{"key": "dc.contributor.advisor", "value": "Juutinen, Petri", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "P\u00f6rh\u00f6, Hanna", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2024-12-13T11:33:27Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2024-12-13T11:33:27Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2024", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/98999", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "Tiedet\u00e4\u00e4n, ett\u00e4 mill\u00e4 tahansa ympyr\u00e4ll\u00e4 on aina mink\u00e4 tahansa annetun kolmion kanssa yhdenmuotoisen kolmion k\u00e4rkipisteet. T\u00e4m\u00e4 voidaan tehd\u00e4 konstruoimalla kolmion k\u00e4rkipisteiden kautta ympyr\u00e4 ja skaalaamalla saatu ympyr\u00e4 sek\u00e4 sill\u00e4 oleva kolmio oikeaan kokoon. Mutta ent\u00e4 jos ympyr\u00e4n tilalla olisikin jokin muu suljettu k\u00e4yr\u00e4? Esimerkiksi ellipsi tai jokin (s\u00e4\u00e4nn\u00f6llinen) monikulmio. T\u00e4ss\u00e4 tutkielmassa n\u00e4ytet\u00e4\u00e4n, ett\u00e4 k\u00e4yr\u00e4n muodolla ei ole v\u00e4li\u00e4, kunhan kyseess\u00e4 on yksinkertainen suljettu k\u00e4yr\u00e4. T\u00e4ll\u00f6in k\u00e4yr\u00e4lt\u00e4 l\u00f6ytyy annetun kolmion kanssa yhdenmuotoisen kolmion k\u00e4rkipisteet. T\u00e4m\u00e4 saadaan n\u00e4ytetty\u00e4 mille tahansa annetulle kolmiolle. Erityisesti milt\u00e4 tahansa yksinkertaiselta suljetulta k\u00e4yr\u00e4lt\u00e4 l\u00f6ydet\u00e4\u00e4n aina tasasivuinen tai tasakylkinen kolmio. Lis\u00e4ksi ty\u00f6ss\u00e4 tarkastellaan kuinka monta annetun kolmion kanssa yh\ndenmuotoista kolmioita k\u00e4yr\u00e4lt\u00e4 l\u00f6ytyy. Niit\u00e4 on k\u00e4yr\u00e4ll\u00e4 \u00e4\u00e4rett\u00f6m\u00e4sti, sill\u00e4 huolimatta siit\u00e4 kuinka pient\u00e4 k\u00e4yr\u00e4n osaa tarkastellaan, niin sielt\u00e4 silti l\u00f6ytyy yhdenmuotoisen kolmion k\u00e4rkipiste.\nT\u00e4ss\u00e4 tutkielmassa yksinkertaisen suljetun k\u00e4yr\u00e4n ei oleteta olevan mill\u00e4\u00e4n tavalla s\u00e4\u00e4nn\u00f6llinen, mik\u00e4 tuo haasteita todistuksiin. Ei esimerkiksi oleteta, ett\u00e4 k\u00e4yr\u00e4 olisi symmetrinen tai ett\u00e4 k\u00e4yr\u00e4ll\u00e4 olisi tangentti kaikissa pisteiss\u00e4. Siit\u00e4 syyst\u00e4 todistus sille, ett\u00e4 k\u00e4yr\u00e4lt\u00e4 l\u00f6ytyy \u00e4\u00e4rett\u00f6m\u00e4n monta annetun kolmion kanssa yhdenmuotoista kolmiota, on pilkottu muutamaan eri tapaukseen riippuen millainen k\u00e4yr\u00e4 on kyseess\u00e4.\nTutkielman aiheeseen liittyy my\u00f6s kirjoitushetkell\u00e4 avoin ongelma. L\u00f6ytyyk\u00f6 yksinkertaiselta suljetulta k\u00e4yr\u00e4lt\u00e4 aina neli\u00f6? T\u00e4h\u00e4n kysymykseen ei t\u00e4ss\u00e4 tutkielmassa perehdyt\u00e4, mutta se on otettu johdannossa huomioon lyhyesti.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Miia Hakanen (mihakane@jyu.fi) on 2024-12-13T11:33:27Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2024-12-13T11:33:27Z (GMT). No. of bitstreams: 0\n Previous issue date: 2024", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "30", "language": "", "element": "format", "qualifier": "extent", "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": "yhdenmuotoisuus", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "suora yhdenmuotoisuus", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Jordanin k\u00e4yr\u00e4", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "yksinkertainen suljettu k\u00e4yr\u00e4", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Jordanin k\u00e4yr\u00e4 ja yhdenmuotoiset kolmiot", "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-202412137820", "language": null, "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Master\u2019s thesis", "language": "en", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu -tutkielma", "language": "fi", "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": "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.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}]
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