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I am looking for a proof of the following statement, known as Archimedes' Lemma:

If an object is divided into two smaller objects, the center of mass of the compound object lies on the line segment joining the centers of mass of the two smaller objects.

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    @DonAntonio: There is a distinction, but in uniform gravity fields they are the same point, hence often used interchangeably. The COG is the average position of the weight distribution, whereas the COM is the average mass distribution.2012-08-26

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Let $A,B$ be two disjoint sets. Let $m_A = \int_{A} dm$ and $m_B = \int_{B} dm$. Then the center of mass of the combined object ($A \cup B$) is: $\frac{\int_{A\cup B} x \, dm}{m_A+m_B} = \frac{\int_{A} x \, dm + \int_{B} x \, dm}{m_A+m_B} = \frac{m_A}{m_A+m_B} \frac{\int_{A} x \, dm }{m_A} + \frac{m_B}{m_A+m_B} \frac{\int_{B} x \, dm }{m_B}.$ The last value is a convex combination of the center of mass of $A$ and $B$, hence lies on the line joining them.