Geodesic
The Euler Line Bisects the Area of only both the Right Triangles and the Isosceles Triangles
Journal for geometry and graphics, Tome 28 (2024) no. 2, pp. 195-198
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It is known that the Euler line bisects the area of both the right triangles and the isosceles triangles. So, to comply with the title, we have proved that the Euler line of any non-right scalene triangle, doesn't bisect its area, intersects the two largest sides of the obtuse triangle, intersects both the smallest and largest sides of the acute triangle, and doesn't pass through a vertex or a midpoint of a side of the triangle. Also, this proof has led to a proof of the well known fact that the three basic centers, the orthocenter H, the centroid G, and the circumcenter O of any non-right scalene triangle ABC are collinear and HG = 2GO.
Classification : 51M04
Mots-clés : Euler line 
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     journal = {Journal for geometry and graphics},
     pages = {195--198},
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S. Abu-Saymeh. The Euler Line Bisects the Area of only both the Right Triangles and the Isosceles Triangles. Journal for geometry and graphics, Tome 28 (2024) no. 2, pp. 195-198. http://geodesic.mathdoc.fr/item/JGG_2024_28_2_JGG_2024_28_2_a5/