https___divisbyzero.com_2010_04_13_an-euler-line-geogebra-applet_.xml
http___free-os.t-com.hr_kbrlekov_Zadatak2_prvi_razred_osobite_tocke_trokuta.htm_1.xml
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Lately I’ve been thinking a lot about Euler and his many contributions to mathematics. So, just for fun I decided to make a Geogeba applet showing the “Euler Line.” In 1763 Euler proved that three different centers of a triangle—the centroid, the orthocenter, and the circumcenter—are collinear. This line is called the Euler line. The centroid of a triangle is the triangle’s center of mass. It is located at the point of intersection of the three medians of the triangle. Recall that a median joins a vertex to the midpoint of the opposite side. The orthocenter of a triangle is the point of intersection of the three altitudes. An altitude is a line through a vertex, perpendicular to the opposite side. Finally, the circumcenter is the point of intersection of the perpendicular bisectors of the three sides of the triangle. The circumcenter is also the center of the circle that circumscribes the triangle. Incidentally, Euler also proved that the distance between the centroid and the orthocenter is twice the distance between the centroid and the circumcenter. (Also on this line, but not shown in the applet, is the center of the nine-point circle, which happens to be the midpoint between that triangle’s orthocenter and circumcenter.)
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Osobite točke trokuta Za pregled ovih stranica trebate imati java plugin kako biste ispravno mogli pregledavati web stranice. http://java.sun.com/j2se/1.5.0/download.jsp Središte trokutu opisane kružnice: Ova točka je sjecište simetrala stranica trokuta.Svaka točka simetrale stranice trokuta jednako je udaljena od dvaju vrhova trokuta. Sve tri simetrale stranica trokuta sijeku se u jednoj točki koja je jednako udaljena od svih vrhova trokuta. Središte trokutu upisane kružnice: Ova točka je sjecište simetrala kuteva trokuta.Svaka točka simetrale kuta trokuta jednako je udaljena od krakova kuta. Sve tri simetrale kuta trokuta sijeku se u točki koja je jednako udaljena od svih stranica trokuta. Sjecište visina trokuta ili ortocentar: Visina trokuta je dužina kojoj je jedan kraj vrh trokuta, a drugi ortogonalna projekcija tog vrha na pravac na kojem leži nasuprotna stranica. Sjecište visina je ortocentar. Težište trokuta: Težišnica trokuta je dužina kojoj je jedna krajnja točka vrh trokuta,a druga polovište nasuprotne stranice. Sve tri težišnice sijeku se u težištu trokta. Ortocentar, težište i središte trokutu opisane kružnice leže na jednom pravcu kojeg zovemo Eulerov pravac.Euler ga je otkrio 1765. godine. Povratka na početnu stranicu