Euler Line Proof

Date: 11/13/2001 at 23:58:00
From: Natalie Loser
Subject: Euler line

Prove that if the Euler line of a triangle passes through a vertex, 
then the triangle is either right or isosceles.

Date: 11/14/2001 at 03:05:52
From: Doctor Floor
Subject: Re: Euler line

Hi, Natalie,

Thanks for writing.

If (for instance) vertex A lies on the Euler line, then the Euler line 
of ABC is the line through A and the centroid G = the median through 
A. So the midpoint of BC is on the line as well.

Now if the orthocenter H lies on the median through A, this means that 
either:

 - A is not equal to H. Then the median through A is also the
   altitude from A, and thus ABC is an isosceles triangle with A as 
   top.

 - A is equal to H. Then A is a right angle, because AB must be the
   altitude from B to AC, and AC must be the altitude from C to AB.

This proves your statement.

If you need more help, just write back.

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