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How did Euclid construct regular polygons in Book IV of the Elements?

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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.

In Book IV of the Elements, Euclid focuses on the construction of regular polygons within circles using only the restricted tools of a compass and an unmarked straightedge.12

Euclid’s approach to these constructions is grounded in the foundational postulates established in Book I, which limit geometric operations to joining two points with a line, extending a line segment, and drawing a circle with a given center and radius. By applying these basic operations, Euclid demonstrated how to construct various polygons, most notably the pentagon, through an ingenious combination of previously proven theorems.132

While techniques such as bisecting angles allow for the construction of certain regular n-gons, Euclid’s work in Book IV provided the essential framework for these elementary geometric constructions. His systematic method of "problems"—where the goal is to construct a figure with specified properties—serves as a cornerstone of Euclidean geometry, establishing a rigorous logical process that would influence mathematical thought for centuries.214