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,

Thank you for your question!

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/