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### Altitudes and Bisectors of a Triangle

```
Date: 05/25/99 at 13:01:28
From: Hanna
Subject: Height and bisector in triangle

We draw the 3 heights of triangle. Then we connect the meeting points
of the heights with the sides of the triangle and get a new triangle.
Prove: the heights of the first triangle are bisectors in the new
triangle.
```

```
Date: 05/25/99 at 16:38:13
From: Doctor Floor
Subject: Re: Height and bisector in triangle

Dear Hanna,

Let us consider the figure you describe: a triangle ABC, with its
orthic triangle A'B'C' made out of the feet of the altitudes (heights)
of ABC. The three altitudes pass through a common point H, the
orthocenter.

We are to show that A'H, B'H, and C'H are the angle bisectors of
triangle A'B'C'. Since this will only be true if ABC is acute (if ABC
is obtuse, two of the lines A'H, B'H, and C'H become 'external' angle
bisectors), I assume your question is limited to such triangles.

Here's a picture:

As you can see, I have drawn a circle through A'BC'H. This can be done
since angles HA'B and HC'B are 90 degrees, so A'BC'H is cyclic.

This also means that since angles HC'A' and HBA' cut off the same arc
of the circle, angle HC'A' = angle HBA' = 90 degrees - angle C (use
triangle BB'C).

In the same way, but now with a circle circumscribing AC'HB', we can
see that angle HC'B' = angle HAB' = 90 degrees - angle C.

But that means that CC' bisects angle C' in triangle A'B'C'. And in
the same way we find that A' and B' are bisected by AA' and BB',
respectively.

Nice problem!

If you need more help, or if you have a math question again, don't
hesitate to write us back.

Best regards,

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

```Date: 06/06/2002 at 16:01:33
From: Zenon Kulpa
Subject: Altitudes and Bisectors of a Triangle

Hi,

In fact, the construction and all your arguments given later
are valid for obtuse triangles as well, and even your drawing
is good for that case: just take the original triangle
to be ABH (instead of ABC) and then the orthocenter will be at C,
while the orthic triangle will remain the same...

Best regards,

-- Zenon
```

```Date: 06/06/2002 at 16:01:26
From: Doctor Floor
Subject: Re: Altitudes and Bisectors of a Triangle

Hi, Zenon,

You are right. When I read the question, I was just thinking of
internal angle bisectors. But indeed in the case of an obtuse
triangle the internal angle bisectors become external ones, but still
angle bisectors. Thanks for the feedback.

Best regards,
- Doctor Floor, The Math Forum
http://mathforum.org/dr.math/ ```
Associated Topics:
High School Geometry
High School Triangles and Other Polygons

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