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?
$\endgroup$ 33 Answers
$\begingroup$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. :-)
$\endgroup$ 1 $\begingroup$Take isogonal conjugate of orthocenter and you get the circumcenter of that triangle.
$\endgroup$ $\begingroup$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.
