What characteristic of the triangle leads the the existence of the orthocenter

Question

We all know that all three altitudes of a triangle meets in the orthocenter of the triangle. It's a quite classical problem and is proven. However, what I really wanna know is what characteristic of the triangle is the profound for this to happen? E.g: Is this because of the sum of 3 internal angles equals 180? In Non-Euclidean geometry, where sum of 3 internal angles is greater or smaller than 180 degree, does the 3 altitudes meets in a single point? Or is it because of another reason?

Answer 1

The proofs I know all use Euklidean geometry (e.g. the orthocenter is the intersection of the middle orthogonals for a bigger triangle).

In synthetic geometry, one can consider translation planes with an orthogonality relation and the Fano axiom (diagonals of a nondegenerate parallelogram intersect), thus minimally allowing the proof above. One can show that this makes the geometry at least a Pappus plane.

Answer 2

It is perhaps interesting to note that the definition of altitudes is perfectly straightforward for simplexes in higher dimensions, but that already in dimension $3$ the altitudes of a general tetrahedron are not concurrent. For instance for the tetrahedron with vertices $(0,0,0)$, $(1,0,0)$, $(0,1,0)$, $(1,0,1)$, two of the altitudes meet in the origin and two others meet in $(1,0,0)$, but there are no other points of intersection; in general position none of the altitudes will intersect.

Answer 3

Vladimir Arnol'd often emphasized that it was because of the Jacobi identity in Lie algebras. I think he might have meant the algebra so(3) represented as $R^3$ with vector cross-product as the multiplication, but am not sure, and it would be nice to see his remark explained.