The Centroid of a Triangle

Date: 02/25/98 at 00:20:39
From: Pauline Jamry
Subject: WHY is the centroid of any triangle the center of its 
balance?

WHY is the centroid of any triangle the center of its balance? 

I have searched your Web site and have found how to find a centroid 
and that it is the center of a triangle's balance, but not why. Is 
there a formula or theorem?

I know how to find the centroid of a triangle by taking the 
intersection of its medians. There is an easier and much quicker 
algebraic way of finding the centroid of a triangle, but I couldn't 
find that.

I thought the reason why the centroid was the balance point was that 
if you take half of any line of a triangle, i.e., its midpoint, and 
just connect it to the opposite vertex, this median and the other two 
of the triangle will be close enough reason to the truth, but I need 
basis and facts.

I also thought that the algebraic way of finding the centroid would 
help me answer my question, or at least give me a clue. 

Date: 02/25/98 at 08:44:38
From: Doctor Jerry
Subject: Re: WHY is the centroid of any triangle the center of its 
balance?

Hi Pauline,

Let me first say that, for me, the most utterly convincing arguments 
for calculating centroids or, more generally, centers of mass, reside 
in calculus. I'm judging that you are not seeking an answer depending 
upon calculus.

Here is a classical argument as to why the centroid of a triangle is 
2/3 of the way from vertex to midpoint of opposite side. The argument 
has two parts.

    1. Convince yourself that the medians do in fact intersect at a   
       point that is 2/3 of the way from the vertex.  This is a  
       question independent of whether this point is the centroid or  
       balance point. I'm assuming that you can do this.

    2. Now take a triangle and draw one median, from vertex C to 
       side c.  Draw three or four representative lines through  
       the median, parallel to side c.  Each of these lines is  
       bisected by the median.  This is a Euclidean geometry exercise.
       So, thinking about the triangle as made from an infinite number  
       of these parallel lines, it appears that the balance point must  
       be on the median, since it bisects each of these lines. Half  
       the mass (line-by-line) is on one side of the median and half  
       the mass is on the other side of the median. Repeat the  
       argument for another median.  So the centroid must be at the  
       intersection of two medians.

This is not a proof; it is, rather, a heuristic argument designed to 
produce belief.

- Doctor Jerry,  The Math Forum
  Check out our Web site  http://mathforum.org/dr.math/