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Diplomsko delo obravnava Haggejevo transformacijo in s to transformacijo tesno povezano izogonalno transformacijo. V prvem delu spoznamo izogonalno transformacijo ter definicijo in konstrukcijo Haggejeve transformacije v primeru, ko točka P leži zunaj trikotniku očrtane krožnice, in v primeru, ko P leži na njej. Nato ugotovimo povezavo med izogonalno transformacijo in Haggejevo transformacijo ter dokažemo, da vse Haggejeve krožnice potekajo skozi višinsko točko. Pogledamo tudi, kam nam Haggejeva transformacija preslika nekaj značilnih točk trikotnika. Sledi obravnava analitične geometrije ter vpeljava trilinearnih koordinat. Na koncu pokažemo, da se z izogonalno transformacijo vse premice preslikajo v stožnice. Ker pa je Haggejeva transformacija v tesni zvezi z izogonalno, velja isto tudi za njo. In this diploma Thesis we deal with Hagge transformation and also with isogonal transformation which is closely related to it. In the first part of the thesis we get aquainted with definition and construction of Hagge transformation in the case when point P lies outside of the circumcicle of the triangle and when point P lies on it. Then we make the connection between isogonal transformation and Hagge transformation and prove that all Hagge circles intersect the same orthocentre. We also look at where some characteristic points of the triangle are transformed. After this, we deal with analytical geometry and introduce trilinear coordinates. At the end we show that with isogonal transformation all lines are transformed into conic sections. Consequently, we show that the same holds true for Hagge transformation. |
