## Two Centers of Mass of a Quadrilateral

Two natural centers of mass of a quadrilateral are not usually at
the same point.

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### The Center of Equal Masses at the Vertices

Suppose equal masses are placed at the vertices of the quadrilateral.
We can pair up the vertices into two pairs. The center of mass of each pair
is the midpoint of the segment connecting them. So the center of mass of the
4 points is the midpoint of the segment connecting the two midpoints. It is
also the intersection of two segments that we can obtain by connecting midpoints
of pairs, when we group the 4 points into pairs two different ways.

We see this in the figure. The center of mass is on either segment
connecting the midpoints of opposite sides.

### The (Area) Center of Mass of the Region

We think of the quadrilateral as being made of some thin, flat material.
We want to find the center of mass of this object. We will call it the center
of mass of the region, or the area center of mass. It is known that for a triangle
the area center of mass is the same as the vertex center of mass, which is the
centroid. The centroid is the point where the medians come together.

Thus if we break the quadrilateral into two triangles and find the
centroid of each, the center of mass of the quadrilateral is on the segment
connecting the two centroids. If we break up the quadrilateral two different
ways, we get two such segments. The intersection is the center of mass of the
region.

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