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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?

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    $\begingroup$ Euler has quite a few things named after him, so there is no need to deprive Euclid of any honors! :D $\endgroup$ Feb 19 '13 at 7:43
  • $\begingroup$ In the hyperbolic plane, the perpendicular bisectors of the sides of a triangle can be concurrent, have a common perpendicular or be asymptotic. In general, then, there is no orthocenter. $\endgroup$ Feb 19 '13 at 7:53
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    $\begingroup$ Have you heard, that the orthocenter is isogonal conjugate of circumcentre? It does not explain everything about the orthocenter, but for me it does explain a lot (isogonal conjugation contains traces of $z \mapsto z^{-1}$ transformation). $\endgroup$
    – dtldarek
    Feb 19 '13 at 8:11
  • $\begingroup$ Something else that bothers me more is that the existence of the Euler line: the orthocenter, centroid, and circumcenter of any triangle are always collinear. It sounds interesting but doesn't it seem a little bit coincident? I mean I know how to prove it but it doesn't sound natural to me. $\endgroup$ Feb 19 '13 at 15:37
  • $\begingroup$ I remember my Physics teacher once tell me about the meaning of numbers: The real numbers forms a perfect continuous line called the Real axis. However no matter how perfect it is, it only like a crack in the wall, which means it's one dimensional, it's the image of the one dimension space. And then they invent the imaginary numbers, which forms another different axis from the Real ones. Together they form the 2 dimensions space, which is better. And they have matrix, which can describe a fully n-dimension space (we all know that a 2x2 matrix is enough to display a Complex number). $\endgroup$ Feb 19 '13 at 15:53
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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.

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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.

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  • $\begingroup$ What is synthetic geometry? $\endgroup$
    – jimjim
    Feb 20 '13 at 2:57
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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.

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