How does Book II of the Elements use geometry to represent algebraic identities?

Book II of Euclid’s Elements is frequently referred to as geometric algebra because it represents algebraic identities as theorems concerning equivalent geometric figures, such as squares, rectangles, and triangles.12

Rather than using modern symbolic notation, Euclid demonstrates these relationships through geometric constructions. For example, the book establishes equalities that correspond to algebraic identities by showing how areas of specific shapes relate to one another. Additionally, Book II includes the construction of the "section"—the division of a line into two parts such that the ratio of the larger segment to the smaller is equal to the ratio of the original line to the larger segment—a concept later known as the golden section. Book II also generalizes the Pythagorean theorem to arbitrary triangles, a result equivalent to the law of cosines.12