Centroid - Center of Gravity

Date: 03/25/2002 at 13:52:54
From: Jack Grahl
Subject: Unique centre of gravity of a triangle

Dear Dr(s). Math,

I would like you to help me understand a result I have been puzzling 
over:

In a problem involving triangles (actually a problem involving cutting 
an irregular shape into two), I realised I had a solution if there was 
a point on a triangle through which any line divided the area of the 
triangle into two equal areas. My expectation was that there must be 
such a point, as in physics a long time ago I learned that everything 
has a centre of gravity. 

However, I tried to prove that any triangle has a (unique) centre 
point like this. I remembered (vaguely) from geometry that the 
triangle has a point called the centre of gravity, and realised that 
it had to lie on each of the lines between a vertex and the mid way 
point of the opposite side. These are easy to construct and all meet 
at a point. I tried to prove that any line through this point cut the 
triangle into two shapes of equal area, but wasn't able to. I also did 
some accurate constructions where it didn't look as if what I was 
trying to prove was right.

The next day I tried another approach, and managed to prove that the 
point where these three lines meet is 2/3 of the way along each of 
them. This means that lines through this point, parallel to a side, 
divide the triangle into a similar triangle, with 4/9 of the area, 
and a trapezium, with 5/9. I'm pretty sure of my proof, and I've also 
found information about the centroid, or "centre of gravity" of a 
triangle, which confirmed it lies 2/3 of the way along each line. 
However, although this point is often described as a "Centre of 
Gravity", I can't find anything that shows it is a centre in the way 
I understand it, nor anything which explains that it isn't.

My problem, in brief, is that I believed everything to have a single 
point, about which the area (or, in physics, mass), can be divided in 
2 IN ANY DIRECTION, and that I seem to have shown that a triangle 
can't have one. I don't understand how, if this is so, physics can 
cope with an object like a flat, solid triangle, or how calculating a 
volume of a solid of rotation can work. Hopefully someone has time to 
deal with a question like this, but I currently don't study maths 
(except by myself) and so don't have a teacher I can ask. 
 
Thanks a lot, Jack

Date: 03/25/2002 at 17:11:29
From: Doctor Rick
Subject: Re: Unique centre of gravity of a triangle

Hi, Jack.

You have made many correct observations. The point you are considering 
is the centroid of a triangle: the intersection of the medians, the 
lines that join each vertex to the midpoint of the opposite side. The 
centroid is a unique point, and each median does divide the triangle 
into two triangles of equal area, but it is not true that ANY line 
through the centroid divides the triangle into regions of equal area.

However, the centroid IS the center of gravity of the triangle. How 
can this be? It's simple: the center of gravity of a 2-dimensional 
figure is NOT defined as a point such that any straight line through 
the point divides the figure into two regions of equal area. Rather, 
if you rest the figure on any straight edge passing through the center 
of gravity, it will balance. That's not the same thing.

The difference is that, when you balance the figure on an edge, parts 
of the figure farther from the fulcrum count more. If you've had 
physics, we are talking about "torque": the product of force times 
distance. The torque exerted by a small part of the figure is the mass 
(which we're assuming is proportional to its area) times the distance 
from the fulcrum (the edge on which we're balancing the figure). The 
figure balances if the sum of the torques from all the parts of the 
figure (counting parts on the left as positive and parts on the right 
as negative) is zero.

If the figure is a triangle and the fulcrum lies along a median, then 
the parts on the two sides of the fulcrum are both triangles, and the 
point farthest from the fulcrum is the same distance on each side. You 
can convince yourself that, if you take thin strips parallel to the 
fulcrum, the same distance from the fulcrum on each side, they are of 
equal length; therefore both the area and the distance are equal, and 
the torques are equal and opposite. Thus each thin strip balances the 
strip on the other side. Altogether, the strips on the left side 
balance the strips on the right side, and the triangle balances.

It's different when you set the fulcrum along a line through the 
centroid that is not a median. In your example where the line is 
parallel to one side of the triangle, that side has 5/9 of the mass - 
but it doesn't stretch out as far from the fulcrum, so its mass 
doesn't count as much as the smaller mass on the other side. It turns 
out that, indeed, the triangle will balance on this fulcrum.

I've just been waving my hands in describing how it works, but it can 
be made quantitative and rigorous. A few years ago in response to a 
Geometry Problem of the Week, I wrote up a fairly rigorous proof 
concerning the center of gravity of triangles and quadrilaterals. If 
you're interested, I'll try to dig it up. It won't exactly answer your 
questions, but it will be sufficiently related that it should be of 
interest.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/   

Date: 03/27/2002 at 12:48:36
From: Jack Grahl
Subject: Unique centre of gravity of a triangle

Dear Dr. Rick,

Thanks a lot for your answer. I think I now have a totally clear 
picture of what a centre of gravity is. Although if I had to define 
it, it would be something like this: A centre of gravity is a point 
on a figure through which any line has the following property: that 
the areas of each shape on either side of the line, multiplied by the 
respective distances between the centre of gravity of that shape and 
the line, are equal to one another.

What you might call a recursive definition, but don't worry, it makes 
sense to me. I've been trying to get more information on working them 
out for other shapes, by building together triangles. Your answer 
suggested to me a method for finding the centre of a quadilateral, 
where, instead of dividing the shape into two pairs of triangles, you 
work with only one pair.

After finding the centroids of the two triangles, and drawing a line 
through it, shouldn't you be able to use their areas (if you know 
them) to work out the ratio between the distances of each centroid to 
the centre of gravity? Then it's simply a case of dividing the line 
segment between them in this ratio to find the point.

Anyway, what I wanted to say is that I'd love to see your proof, if 
you still have it.

Thanks a lot for your help,
Jack Grahl

Date: 03/27/2002 at 13:32:21
From: Doctor Rick
Subject: Re: Unique centre of gravity of a triangle

Hi, Jack.

Yes, that's a way to find the center of gravity of a quadrilateral. 
The GeoPoW problem was just this, and I stated that solution, but I 
found another method simpler geometrically. You can draw both 
diagonals of the quadrilateral, connect the centroids of the two 
triangles into which one diagonal divides the quadrilateral, and do 
the same for the other diagonal. The intersection of the two lines is 
the center of gravity of the quadrilateral.

The tough part was to prove, to my satisfaction, that this really does 
the job. The approach I came up with is essentially the same as your 
recursive definition of the center of gravity. Annie Fetter, the 
moderator, thought I was being too picky. Maybe she's right, I don't 
know ...

The problem is found here:

  The Center of Gravity of a Quadrilateral, July 14-18, 1997
  http://mathforum.org/geopow/archive/19970718.geopow.html   

Here is what I wrote:

PROBLEM: Find the center of gravity of a quadrilateral. 
Construction: 

Given a quadrilateral ABCD, find the midpoint of each side (the 
midpoints of AB, BC, CD, and DA are E, F, G, and H respectively). 

Take two opposite vertices (A and C) and draw lines connecting each to 
the midpoints of the two sides that meet at the opposite vertex: AF, 
AG, CE, and CH. 

Label the intersections of AF and CE, P, and of AG and CH, Q. Connect 
them. 

Repeat with the other two vertices (B and D); label the intersections 
of BH and DE, R, and of BG and DF, S. 

The intersection of PQ and RS is the centroid, O. 

Discussion: 

A tough problem, unless I'm missing something obvious! I found the 
construction fairly easily, but I had to use what I remember about 
centers of gravity from physics. Bringing what I know into the context 
of geometry is the hard part, starting with finding a useable 
geometrical definition of the center of gravity (centroid) of an 
arbritrary figure. 

First, the bonus: Prove that the six small triangles between the 
medians have the same area. 

PROOF: Consider triangle ABC in my figure. 

             C
            /\
           / \ \
          /      \
         /    \    \
        /            \
     F /       \       \ G
      / \              / \
     /      \   \  /       \
    /          / \           \
   /       /     \    \        \
  /    /                  \      \
 /________________\________________\
A                 E                 B

The areas of triangles CAE and CEB are equal because they have the 
same base (AE = EB) and height (distance of C from AB). Each is 
therefore half the area of triangle ABC. 

Likewise triangles ABF and AFC have the same area, again half the area 
of triangle ABC. 

Thus the areas of ABF and CEB are equal, and if we subtract the common 
area BFPE, we have the areas of PAE and PFC equal. 

Again, triangles PAE and PEB have the same area because their bases 
and height are equal, and likewise for PBF and PFC. 

So we have PEB = PAE = PFC = PBF, and in a similar fashion the 
remaining two small triangles can be shown to be equal to the others. 
The implication of the problem statement is that the definition of a 
centroid has something to do with this fact, that is, lines passing 
through the centroid dividing a figure into equal areas. But this is 
not the case. The centroid is the mean (average) location of all the 
points contained in a figure. Any line passing through the centroid 
can serve as a pivot and the figure will "balance" (supposing it is 
cut out of wood, for instance, of uniform thickness). In physics, this 
means the "moments" on both sides are equal (and opposite). It does 
not necessarily mean the areas on both sides are the same; a smaller 
area farther from the pivot can balance a larger area closer to the 
pivot. 

In the case of the medians of a triangle, the figures on both sides 
are triangles; therefore the area is distributed the same way, so 
equal moments imply equal areas. Other lines through the centroid give 
a triangle on one side and a quadrilateral on the other; the mass 
distribution is different - the quadrilateral has more mass farther 
out - so the triangle needs a larger area to balance it.

Now I will try to put the centroid of a general figure in a classical- 
geometry context. I define the centroid by three postulates:

1. The centroid of a figure with a rotational symmetry is the rotation 
axis.

2. If two figures a and b are similar, their centroids A and B form 
similar figures with corresponding elements of a and b.

3. Given two figures a and b with centroids A and B, the centroid of 
the two figures taken together is the point C on the line AB, such 
that AC * Area(a) = CB * Area(b).

I don't know if these are all necessary or sufficient, but they are 
consistent with the definition of a centroid as the mean location of 
the points in a figure, and they are sufficient for my purpose. I have 
one thing to prove before I can justify my construction: that this 
definition is consistent with the centroid we know.

THEOREM: The centroid of a triangle is the intersection of its 
medians. PROOF: Given any triangle ABC, suppose its centroid is P. 
Construct the midpoints of AB, BC, and CA at D, E, and F respectively. 
Triangle ABC can be divided into parallelogram BDFE and triangles ADF 
and FEC. Call the centroids of these figures Q, R, and S respectively.

Triangles ADF and FEC are congruent, so they have the same area, 
therefore by (1) above, the centroid T of the two triangles taken 
together is at the midpoint of RS.

The centroid Q of parallelogram BDFE is on the diagonal BF by (1) 
above.

Triangle ADF is similar to triangle ABC. Therefore by (2) above, 
triangle ARF is similar to APC; angle RAF = angle PAC, so R lies on 
line AP; and since F is the midpoint of AC, R is the midpoint of AP. 
Similarly, S is the midpoint of CP.

Triangle RPS is similar to triangle APC. Therefore T, the midpoint of 
RS, lies on PF.

Triangle ABC is the sum of parallelogram BDFE (with centroid Q) and 
triangles ADF and FEC (taken together, with centroid T), so by (3) 
above, the centroid P of ABC is on the line QT.

Since Since P, T, and F are collinear by (4), and P, T, and Q are 
collinear by (5), then P, T, F, and Q are all collinear. Since Q, B, 
and F are collinear by (2), we can add B to the list, so in 
particular, P is on BF. But BF is a median of triangle ABC.

By similar arguments, P is on the other two medians of ABC as well. In 
other words, the centroid of any triangle ABC is the intersection of 
its medians.

Now I am finally ready to justify my construction of the centroid of a 
quadrilateral. A quadrilateral can be divided into two triangles in 
two ways: by cutting along either diagonal. If I make one such cut, I 
know the centroid lies along the line joining the centroids of the two 
triangles (P and Q), by definition part (3).

I could at this point find the centroid by choosing O on PQ such that 
OP:OQ = Area(ACD):Area(ABC). But it is much easier with compass and 
straightedge to take advantage of the other pair of triangles and find 
another line (RS) on which the centroid lies. The centroid must of 
course lie at the intersection of the two lines.

- Doctor Rick, The Math Forum 
  http://mathforum.org/dr.math/