essay understanding - Archimedes exhaustion

In this article:
https://www.maa.org/external_archive/joma/Volume7/Aktumen/Polygon.html
The writer uses the law of sines.
I don't understand why he uses it and what purpose for?

I wouldn't have called it the law of sines; what is being used, twice, is the area formula, area = 1/2 b c sin(A). What I call the law of sines is never mentioned in the article!

If you take the link provided the first time they mention the law of sines, you see that the Wikipedia article proves the law of sines from this area formula; so perhaps the author considers them to be equivalent.
 
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