Implementing Euclid’s straightedge and compass constructions in type theory

  • Ariel Kellison
  • Mark Bickford
  • Robert Constable


Constructions are central to the methodology of geometry presented in the Elements. This theory therefore poses a unique challenge to those concerned with the practice of constructive mathematics: can the Elements be faithfully captured in a modern constructive framework? In this paper, we outline our implementation of Euclidean geometry based on straightedge and compass constructions in the intuitionistic type theory of the Nuprl proof assistant. A result of our intuitionistic treatment of Euclidean geometry is a proof of the second proposition from Book I of the Elements in its full generality; a result that differs from other formally constructive accounts of Euclidean geometry. Our formalization of the straightedge and compass utilizes a predicate for orientation, which enables a concise and intuitive expression of Euclid’s constructions.


Constructive type theory Constructive geometry Foundations of geometry 

Mathematics Subject Classification (2010)

68T15 03B35 03F55 


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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Ariel Kellison
    • 1
  • Mark Bickford
    • 1
  • Robert Constable
    • 1
  1. 1.Department of Computer ScienceCornell UniversityIthacaUSA

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