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How does Euclid’s third postulate allow a circle to be constructed?

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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.

Euclid’s third postulate states that a circle can be constructed when a point for its center and a distance for its radius are given.1

This postulate serves as a fundamental principle for geometric constructions within the Elements. By defining the requirements for a circle—a specific center point and a fixed distance—it establishes the mathematical basis for using a compass to draw circles in Euclidean geometry. Along with the first two postulates (which allow for the drawing of straight lines and the extension of line segments), the third postulate restricts classical geometric constructions to the use of the two standard Euclidean tools: the compass and the unmarked straightedge.2