How does Euclid’s work in Book II relate to the law of cosines?

In Book II of the Elements, Euclid presents propositions that serve as the geometric foundation for what is now known as the law of cosines.12

While Euclid did not use modern trigonometric notation or the concept of the cosine function, he addressed the relationships between the sides of both acute and obtuse triangles using geometric constructions. Book II is often described as geometric algebra because it expresses algebraic identities through the properties of equivalent geometric figures. In this context, Euclid generalized the Pythagorean theorem to apply to arbitrary triangles, providing a result that is mathematically equivalent to the modern law of cosines formula:13

$$c^2=a^2+b^2-2ab\cos C$$

By proving these relationships through geometry rather than trigonometry, Euclid established the fundamental principles that would later be formalized by mathematicians such as Jamshīd al-Kāshī.12