## Abstract

We used computer proof-checking methods to verify the correctness of our proofs of the propositions in Euclid Book I. We used axioms as close as possible to those of Euclid, in a language closely related to that used in Tarski’s formal geometry. We used proofs as close as possible to those given by Euclid, but filling Euclid’s gaps and correcting errors. Euclid Book I has 48 propositions; we proved 235 theorems. The extras were partly “Book Zero”, preliminaries of a very fundamental nature, partly propositions that Euclid omitted but were used implicitly, partly advanced theorems that we found necessary to fill Euclid’s gaps, and partly just variants of Euclid’s propositions. We wrote these proofs in a simple fragment of first-order logic corresponding to Euclid’s logic, debugged them using a custom software tool, and then checked them in the well-known and trusted proof checkers HOL Light and Coq.

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## Appendices

### Appendix A: Formal proof of Prop. I.1

The reader may compare the following proof to Euclid’s. The conclusion ELABC, that *ABC* is equilateral, is reached about halfway through, and that corresponds to the end of Euclid’s proof. The firsts half of our proof corresponds fairly naturally to Euclid’s, except for quoting the circle-circle axiom, and verifying that is hypotheses are satisfied.

The last half of our proof is devoted to proving that *ABC* is a triangle, that is, the three points are not collinear. Note the use of lemma partnotequalwhole. If Euclid had noticed the need to prove that *ABC* actually is a triangle, he would have justified it using the common notions, applied to equality (congruence) of lines. This version of “the part is not equal to the whole” is not an axiom for us, but a theorem.

At the request of the referee we present this proof in a typeset form rather than in its native Polish form. Obviously further mechanical processing can increase its superficial resemblance to Euclid’s style, but the point of our present work is simply its mechanically-checked *correctness*.

###
**Proposition A.1** (Prop. I.1)

*On a given finite straight line to construct an equilateral*
*triangle.*

### Proof

Let *J* be such that *J* is the circle of center *A* and radius *A**B* by postulate Euclid3.

*B* ≠ *A* by lemma inequalitysymmetric.

Let *K* be such that *K* is the circle of center *B* and radius *B**A* by postulate Euclid3.

Let *D* be such that *A* is strictly between *B* and *D* ∧ *A**D*≅*A**B* by lemma localextension.

*A**D*≅*B**A* by lemma congruenceflip.

*B**A*≅*B**A* by common notion congruencereflexive.

*D* is outside circle *K* by definition of outside.

*B* = *B* by common notion equalityreflexive.

*B* is inside circle *K* by definition of inside.

*A**B*≅*A**B* by common notion congruencereflexive.

*B* is on circle *J* by definition of on.

*D* is on circle *J* by definition of on.

*A* = *A* by common notion equalityreflexive.

*A* is inside circle *J* by definition of inside.

Let*C* be such that *C* is on circle *K* ∧ *C* is on circle *J* by postulate circle-circle.

*A**C*≅*A**B* by axiom circle-center-radius.

*A**B*≅*A**C* by lemma congruencesymmetric.

*B**C*≅*B**A* by axiom circle-center-radius.

*B**C*≅*A**B* by lemma congruenceflip.

*B**C*≅*A**C* by lemma congruencetransitive.

*A**B*≅*B**C* by lemma congruencesymmetric.

*A**C*≅*C**A* by common notion equalityreverse.

*B**C*≅*C**A* by lemma congruencetransitive.

*A**B**C* is equilateral by definition of equilateral.

*B* ≠ *C* by axiom nocollapse.

*C* ≠ *A* by axiom nocollapse.

Let show that *C* is strictly between *A* and *B* does not hold by contradiction:

{

\({A}{C} \ncong {A}{B}\) by lemma partnotequalwhole.

*C**A*≅*A**C* by common notion equalityreverse.

*C**A*≅*A**B* by lemma congruencetransitive.

*A**C*≅*C**A* by common notion equalityreverse.

*A**C*≅*A**B* by lemma congruencetransitive.

We have a contradiction.

}

Let show that *B* is strictly between *A* and *C* does not hold by contradiction:

{

\({A}{B} \ncong {A}{C}\) by lemma partnotequalwhole.

*A**B*≅*C**A* by lemma congruencetransitive.

*C**A*≅*A**C* by common notion equalityreverse.

*A**B*≅*A**C* by lemma congruencetransitive.

We have a contradiction.

}

Let show that *A* is strictly between *B* and *C* does not hold by contradiction:

{

\({B}{A} \ncong {B}{C}\) by lemma partnotequalwhole.

*B**A*≅*A**B* by common notion equalityreverse.

*B**A*≅*B**C* by lemma congruencetransitive.

We have a contradiction.

}

Let show that *A**B**C* are collinear does not hold by contradiction:

{

*A* ≠ *C* by lemma inequalitysymmetric.

*A* = *B*∨*A* = *C*∨*B* = *C*∨*A* is strictly between *B* and *C*∨*B* is strictly between *B* and *C*∨*C* is strictly between *A* and *B* by definition of collinear.

We have a contradiction.

}

*A**B**C* is a triangle by definition of triangle. □

### Appendix B: Axioms and definitions

The following formulas are presented in a format that can be cut and pasted, even from a pdf file.

### 1.1 Definitions

*A and B are distinct points*

*A, B, and C are collinear*

*A, B, and C are not collinear*

*P is inside (some) circle J of center C and radius AB*

*P is outside (some) circle J of center U and radius VW*

*B is on (some) circle J of center U and radius XY*

*ABC is equilateral*

*ABC is a triangle*

*C lies on ray AB*

*AB is less than CD*

*B is the midpoint of AC*

*Angle ABC is equal to angle abc*

*DBF is a supplement of ABC*

*ABC is a right angle*

*PQ is perpendicular to AB at C and NCABP*

*PQ is perpendicular to AB*

*P is in the interior of angle ABC*

*P and Q are on opposite sides of AB*

*P and Q are on the same side of AB*

*ABC is isosceles with base BC*

*AB cuts CD in E*

*Triangle ABC is congruent to abc*

*Angle ABC is less than angle DEF*

*AB and CD are together greater than EF*

*AB, CD are together greater than EF,GH*

*ABC and DEF make together two right angles*

*AB meets CD*

*AB crosses CD*

*AB and CD are Tarski parallel*

*AB and CD are parallel*

*ABC and DEF are together equal to PQR*

*ABCD is a parallelogram*

*ABCD is a square*

*ABCD is a rectangle*

*ABCD and abcd are congruent rectangles*

*ABCD and abcd are equal rectangles*

*ABCD is a base rectangle of triangle BCE*

*ABC and abc are equal triangles*

*ABCD and abcd are equal quadrilaterals*

### 1.2 Common notions

### 1.3 Axioms of betweenness and congruence

### 1.4 Postulates

### 1.5 Axioms for equal figures

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Beeson, M., Narboux, J. & Wiedijk, F. Proof-checking Euclid.
*Ann Math Artif Intell* **85**, 213–257 (2019). https://doi.org/10.1007/s10472-018-9606-x

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DOI: https://doi.org/10.1007/s10472-018-9606-x