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### Characteristics of an Orthocenter

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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.

```

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

Hi, Diego,

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/
```
Associated Topics:
High School Geometry
High School Triangles and Other Polygons
Middle School Geometry
Middle School Triangles and Other Polygons

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