## 6 Citations

### A Direct Proof of the Steiner–Lehmus Theorem

- Mathematics, PhilosophyThe Mathematical Intelligencer
- 2021

I n his article ‘‘An Elementary-Minded Mathematician’’ [1], which appeared in the summer 2021 issue of this journal, O. A. S. Karamzadeh mentions, regarding the possibility of a direct proof of the…

### Lippmann’s axiom and Lebesgue’s axiom are equivalent to the Lotschnittaxiom

- MathematicsBeiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry
- 2019

We prove that both Lippmann’s axiom of 1906, stating that for any circle there exists a triangle circumscribing it, and Lebesgue’s axiom of 1936, stating that for every quadrilateral there exists a…

### Lippmann’s axiom and Lebesgue’s axiom are equivalent to the Lotschnittaxiom

- MathematicsBeiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry
- 2019

We prove that both Lippmann’s axiom of 1906, stating that for any circle there exists a triangle circumscribing it, and Lebesgue’s axiom of 1936, stating that for every quadrilateral there exists a…

### A machine-checked direct proof of the Steiner-lehmus theorem

- MathematicsCPP
- 2022

This paper has formalized a constructive axiom set for Euclidean plane geometry in a proof assistant that implements a constructive logic and has built the proof of the Steiner-Lehmus theorem on this constructive foundation.

### Implementing Euclid’s straightedge and compass constructions in type theory

- MathematicsAnnals of Mathematics and Artificial Intelligence
- 2018

This paper outlines the implementation of Euclidean geometry based on straightedge and compass constructions in the intuitionistic type theory of the Nuprl proof assistant, which enables a concise and intuitive expression of Euclid’s constructions.

### Parallel Postulates and Continuity Axioms: A Mechanized Study in Intuitionistic Logic Using Coq

- MathematicsJournal of Automated Reasoning
- 2017

This paper develops a large library in planar neutral geometry, including the formalization of the concept of sum of angles and the proof of the Saccheri–Legendre theorem, which states that assuming Archimedes’ axiom, the sum of the angles in a triangle is at most two right angles.

## References

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This paper completely settle the questions about implications between the three versions of the parallel postulates: the strong parallel postulate easily implies Euclid 5, and in fact Euclid5 also implies the strong Parallel Postulate, although the proof is lengthy, depending on the verification that Euclid 4 suffices to define multiplication geometrically.

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It is shown that if ECG proves an existential theorem, then the object proved to exist can be constructed from parameters, using the basic constructions of ECG (which correspond to the Euclidean straightedge-and-compass constructions).

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A methodology for the automated preparation and checking of the input files for the theorems in Tarskian geometry, to ensure that no human error has corrupted the formal development of an entire theory as embodied in two hundred input files and proofs.

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A classical theory of Desarguesian geometry, originating with D. Hilbert in his 1899 treatise Grundlagen der Geometrie, leads from axioms to the construction of a division ring from which coordinates…

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- PhilosophyAnn. Pure Appl. Log.
- 1998

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- PhilosophyBulletin of Symbolic Logic
- 1999

Abstract This paper is an edited form of a letter written by the two authors (in the name of Tarski) to Wolfram Schwabhäuser around 1978. It contains extended remarks about Tarski's system of…

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- Mathematics
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The classical theory of plane projective geometry is examined constructively, using both synthetic and analytic methods. The topics include Desargues’s Theorem, harmonic conjugates, projectivities,…