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How does the first Euclidean postulate describe a line through two points?

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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 Euclidean geometry, the first postulate establishes the fundamental relationship between points and lines. Stated in modern terms, it asserts that given any two points, there is a straight line that joins them.1

While Euclid’s original formulation provided this intuitive basis, the mathematician David Hilbert later refined the axiom to specify that for any two distinct points, there exists a line containing them and that this line is unique. This principle serves as a foundational axiom for constructing geometric figures and deriving further theorems within the system.1