Characteristics of an Orthocenter

Date: 11/12/1999 at 03:08:00
From: Diego Carbonetti
Subject: Uses of an orthocenter

I need to know some characteristics of the orthocenter of a triangle, 
other than the fact that it lies on the Euler line.

Thank you for your time.

Date: 11/12/1999 at 09:56:22
From: Doctor Floor
Subject: Re: Uses of an orthocenter

Hi, Diego,

Thanks for your question.

Here are some nice properties of the orthocenter H of triangle ABC:

1. Pick three points out of the four points A, B, C, and H. The fourth 
   point is the orthocenter of the first three. For this reason the 
   four points are called an orthocentric system. One of the famous 
   orthocentric systems is formed by the incenter and the three 
   excenters.

2. Reflect H about the sidelines of ABC. The three reflections lie on 
   the circumcircle of ABC.

3. There are four triangles you can make out of the four points A, B, 
   C, and H. All four triangles have congruent circumcircles.

4. Let A' be the foot of the altitude AH. And let B' and C' be 
   constructed likewise. A'B'C' is called the orthic triangle of ABC. 
   When ABC is acute, then the incenter of A'B'C' is H; when it is 
   obtuse, then the incenter of A'B'C' is the vertex of the obtuse 
   angle.

If you need more help, just write us back.

Best regards,
- Doctor Floor, The Math Forum
  http://mathforum.org/dr.math/