Show that Propositions II 12 and 13 are essentially the law of cosin.pdf

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Propositions II 12 and 13 from Euclid's Elements are equivalent to the law of cosines, which relates the lengths of all three sides of a triangle to the cosine of one of its angles. The document shows that the law of cosines can be proven through consideration of three cases for the angle C: when C is obtuse, a right angle, or acute. This verifies that Propositions II 12 and 13 are essentially the same as the law of cosines.

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Show that Propositions II 12 and 13 are essentially the law of cosines
Solution
The law of cosines c2 = a2 + b2 – 2ab cos C There are two other versions of the
law of cosines, a2 = b2 + c2 – 2bc cos A and b2 = a2 + c2 – 2ac cos B. Since the three verions
differ only in the labelling of the triangle, it is enough to verify one just one of them. We'll
consider the version stated first. In order to see why these laws are valid, we'll have to look at
three cases. For case 1, we'll take the angle C to be obtuse. In case 2, angle C will be a right
angle. In case 3, angle C will be acute.

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