Journal for Geometry and Graphics 28 (2024), No. 2, 195--198
Copyright by the authors licensed under CC BY SA 4.0
The Euler Line Bisects the Area of only both the Right Triangles and the Isosceles Triangles
Sadi Abu-Saymeh
Concord, North Carolina, U.S.A.
ssaymeh@yahoo.com
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.
Keywords: Euler line.
MSC: 51M04.
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