Incenter, Orthocenter, Circumcenter, Centroid

Date: 01/05/97 at 21:25:17
From: Kristy Beck
Subject: Euler line

I have been having trouble finding the Euler line on a triangle. 
If you would explain to me, I would be most grateful!

Thank you,
Kristy Beck

Date: 01/06/97 at 01:51:53
From: Doctor Pete
Subject: Re: Euler line

The Euler line of a triangle is the line which passes through the 
orthocenter, circumcenter, and centroid of the triangle.  

       


The orthocenter is the intersection of the triangle's altitudes.

       


The circumcenter is the center of the circumscribed circle (the 
intersection of the perpendicular bisectors of the three sides).

       


The centroid is the intersection of the three medians of the triangle.

       


There's also the incenter, which is the intersection of the angle 
bisectors of the triangle.

       

-Doctor Pete,  The Math Forum

NEWSGROUP DISCUSSIONS

For proofs involving the center of gravity and the circumcenter of a
triangle, see Eileen Steven's question sent to the newsgroup
geometry-pre-college, and Eileen Schoaff's answer:

  http://mathforum.org/kb/message.jspa?messageID=1077584   

  http://mathforum.org/kb/thread.jspa?forumID=128&threadID=351544   


For some properties of the orthocenter of a triangle, see Michael 
Keyton's posting to the newsgroup geometry-puzzles: 

  http://mathforum.org/kb/message.jspa?messageID=1084283   
  
  
ON THE WEB: TRIANGLE CENTERS
  
Prof. Clark Kimberling's page of Triangle Centers provides a listing, 
with links to descriptions and illustrations, of 20th-century triangle 
centers, including: Schiffler Point, Exeter Point, Parry Point, 
congruent isoscelizers point, Yff Center of Congruence, isoperimetric 
point and equal detour point, Ajima-Malfatti Points, Apollonius Point, 
Morley Centers, Hofstadter Points, and equal parallelians points; and 
some classical triangle centers: centroid, incenter, circumcenter, 
orthocenter, Fermat Point, nine-point center, symmedian (or Lemoine) 
point, Gergonne point, Nagel point, Mittenpunkt, Spieker center, 
Feuerbach point, isodynamic points, and Napoleon points.

  http://faculty.evansville.edu/ck6/encyclopedia/   

-Doctor Sarah,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/