Does the orthocenter have any special properties?

Question

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?

Then a Google search should work, and sites like Mathworld or Wikipedia and their sources might help. — Sawarnik, Oct 24 '13 at 09:27

Jean-Claude Arbaut · Answer 1 · 2013-03-15T15:48:44.343

1

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. :-)

edited Mar 15 '13 at 15:48

answered Mar 15 '13 at 15:36

Jean-Claude Arbaut

23,277

Please take a look on the following question: https://math.stackexchange.com/questions/2321816/proving-lines-are-perpendicular – Blind Jun 15 '17 at 21:13

score 0 · Answer 2 · answered Oct 03 '15 at 19:16

0

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

answered Oct 03 '15 at 19:16

enedil

1,710

score 0 · Answer 3 · answered Mar 02 '17 at 15:33

When constructing the orthocenter or triangle T, the 3 feet of the altitudes can be connected to form what is called the orthic triangle, t. When T is acute, the orthocenter is the incenter of the incircle of t while the vertices of T are the excenters of the excircles of t. When the triangle is obtuse then the roles of the vertex of the obtuse angle and the orthocenter are reversed.

Does the orthocenter have any special properties?

3 Answers3