TITLE: In which geometries do triangles have an Euler line?
QUESTION [11 upvotes]: In Euclidean geometry, the centroid, orthocenter and circumcenter of a triangle lie on a line.
In which other geometries does this hold?

REPLY [6 votes]: As shown by Richard Baldus in 1929 (Ueber Eulers Dreieckssatz in der absoluten Geometrie, Sitzungsberichte  der  Heidelberger  Akademie  der Wissenschaften. Mathematisch-naturwissenschaftliche  Klasse  11. Abhandlung), the Euler line exists in non-Euclidean geometry only in the case of isosceles triangles. A different proof for that result and a related one for Feuerbach's circle can be found in H. Kuenneth, H.: Der Schwerpunkt, die Eulersche Gerade und der Feuerbachsche Kreis in der absoluten Geometrie.
Jahresber. Deutsch. Math.-Verein. 43 (1933), 65–77.