## Proposition 12

This proposition for obtuse angles, together with the next one for acute triangles, complement the Pythagorean theorem, Proposition I.47, for right triangles. Prop.I.47 says that if triangle ABC has a right angle at A, then

a2 = b2 + c2

where a, b, and c are the sides opposite the angles A, B, and C, respectively.
 In this proposition, II.12, the angle A is obtuse rather than right, and the conclusion is that a2 = b2 + c2 - 2ch where h is the height of the triangle when c is taken as the base of the triangle. The next proposition, II.13 has the same conclusion, but the hypothesis is that the angle at A is acute rather than obtuse.

This conclusion is very close to the law of cosines for oblique triangles.

a2 = b2 + c2 – 2bc cos A,

since the height h equals b cos A. Trigonometry was developed some time after the Elements was written, and the negative numbers needed here (for the cosine of an obtuse angle) were not accepted until long after most of trigonometry was developed. Nonetheless, this proposition and the next may be considered geometric versions of the law of cosines.

Neither this nor the next is used in the rest of the Elements.

Next proposition: II.13

Previous: II.11

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