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Each of the commonly known triangle centers I know has some sort of special property. For example:

  • The incenter is the center of the inscribed circle.
  • The circumcenter is the center of the circle defined by three points.
  • The centroid is the gravitational center of an object.

Does the orthocenter have any similar property? How about the symmedian center or the nine-point center?

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So do you mean properties which are not directly geometric? – Sawarnik Oct 23 '13 at 16:14
I suppose that would work. – Joe Z. Oct 24 '13 at 1:37
Then a Google search should work, and sites like Mathworld or Wikipedia and their sources might help. – Sawarnik Oct 24 '13 at 9:27

Look at Euler line or Euler circle, and these are just examples. There are numerous properties in the triangle, many involving the orthocenter. And there are litterally hundreds of special points. Some even say it's a sin to spend too much time looking for such properties. :-)

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Take isogonal conjugate of orthocenter and you get the circumcenter of that triangle.

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