# Archimedes and center of gravity of an triangle

I want to know: how Archimedes determined the center of gravity of arbitrary triangle? Do you know the rigorous method?

If you know a website or document please tell me. I'm a looking for a reliable reference.

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You might be interested in this. – J. M. Sep 13 '11 at 7:30
@J.M. nice document. Thanks! – Hiperion Sep 13 '11 at 14:44
A related question: Did anyone before Archimedes ever think about the concept of a center of gravity? – Michael Hardy Sep 14 '11 at 14:27
@Michael, I have not heard from anyone. – Hiperion Sep 14 '11 at 22:38

Historical reference: On the Equilibria of Planes, Book I, Propositions 13-14.

Here is an online version (containing comments from Eutocius) translated by Henry Mendell, with pictures included.

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+1. Just for completeness: On the Equilibria of Planes is a work written by Archimedes. The link above is a translation of this work along with the commentary by Eutocius written in the sixth century. – ShreevatsaR Sep 13 '11 at 6:45
There is considerable controversy about the authorship of Book I (at least) of Equilibrium of Planes, largely because the arguments are much below the usual high standards of Archimedes. – André Nicolas Sep 13 '11 at 6:48
@anon, thanks, I'm reading... – Hiperion Sep 13 '11 at 6:58
André Nicolas and ShreevatsaR, good to know. Thank you very much. – Hiperion Sep 13 '11 at 7:01
Huh. I would have imagined Archimedes would know about using plumb lines to reckon a triangle's centroid. – J. M. Sep 13 '11 at 7:30

Although I don't have a historical reference, I'm sure he did it by noting that a line segment from a vertex to the mid-point of the opposite side bisects the triangle into triangles of equal area. So doing this construction from two vertices finds the centroid as the intersection of the constructed lines.

Of course, you can prove that this construction bisects area by the usual area formula $A =\frac{1}{2} bh$

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thank you very much! I'm a looking for a reliable reference too. – Hiperion Sep 13 '11 at 6:32