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### The Centroid of a Triangle

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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.
```

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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.
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/
```
Associated Topics:
High School Geometry
High School Triangles and Other Polygons

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