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How is Euclid’s general result for triangles related to the law of cosines?

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Three theorems of congruent triangles — Euclidean geometry This diagram depicts three basic theorems that triangles are congruent—that is, they are of equal shape and size—if: (1) two sides and the included angle are equal, designated SAS; (2) two angles and the included side are equal, designated ASA; or (3) all three sides are equal, designated SSS.

The law of cosines is a modern generalization of the Pythagorean theorem that relates the lengths of the sides of any triangle. While the modern formula uses trigonometric notation, its geometric foundation is rooted in the work of the Greek mathematician Euclid.12

In his seminal work, Elements, Euclid presented propositions that are mathematically equivalent to the law of cosines. Because the cosine function had not yet been developed in his time, Euclid expressed these relationships through geometric constructions rather than algebraic equations. His propositions addressed the properties of both acute and obtuse triangles, providing the necessary geometric framework that would later be formalized as the law of cosines by mathematicians such as Jamshīd al-Kāshī in the 15th century.12