Undergrad Prove that only one straight line passes through two point

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SUMMARY

The discussion centers on proving that only one straight line can pass through two distinct points in Euclidean geometry. It emphasizes the necessity of including this principle in the axioms to avoid non-Euclidean scenarios, where multiple lines can intersect two points, such as on a sphere where all longitude lines converge at the poles. The distinction between Euclidean and non-Euclidean geometry is crucial for understanding this concept.

PREREQUISITES
  • Understanding of Euclidean geometry principles
  • Familiarity with axiomatic systems in mathematics
  • Basic knowledge of non-Euclidean geometry
  • Concept of distinct points in geometric space
NEXT STEPS
  • Study the axioms of Euclidean geometry
  • Explore the properties of non-Euclidean geometries, such as spherical and hyperbolic geometry
  • Learn about the implications of different geometric systems on mathematical proofs
  • Investigate the concept of parallel lines and their behavior in various geometries
USEFUL FOR

Mathematicians, geometry students, educators, and anyone interested in the foundational principles of geometric theory.

parshyaa
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I was just thinking of basic definitions of geometry and i came to this question, so how could i prove that only one straight line passes through two distinct points.
 
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You have to require that in the axioms, otherwise you get non-euclidean geometry where multiple different lines can go through two points.
 
Simple example - non-Euclidean: sphere, where all longitude lines go through both poles.
 

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