Which statements are equivalent to the parallel postulate?

Question

I would like to have a long-ish list of statements that are equivalent to the parallel postulate.

If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.

https://en.wikipedia.org/wiki/Parallel_postulate gives 14:

Heath's "Euclid, the thirteen books of the elements" Dover edition mentions some of them, but also4 other ones.

My main question is:

Are there any more?

And related: Is there any publication that gives (many) more of them?

If you know more of them, add them to the comunity wiki answer below, if possible give referrences to where they come from.

(so we get an even longer list :) )

For this question do assume all other axioms of neutral or absolute geometry including continuity https://en.wikipedia.org/wiki/Absolute_geometry (for those who want to be picky).

Answer 1

Here the list organised by main subject.

Add more if you know some, but add reference to where it comes from.

If a proposition falls under more than 2 subjects you may add them under both. Like triangle 5 ( Every triangle can be circumscribed ) and circle 1 ( Given any three points not on a straight line, there exists a circle through them).

Lines:

Euclid: If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles. [1]

1. There is at most one line that can be drawn parallel to another given one through an external point. (Playfair's axiom)[1,6]

2. There exists a pair of straight lines that are at constant distance from each other.[1]

3. Two lines that are parallel to the same line are also parallel to each other.[1,6]

4. If a line intersects one of two parallel lines, both of which are coplanar with the original line, then it also intersects the other. (Proclus' axiom) [1,6]

5. if two straight lines are parallel, they are figures opposite to (or the reflex of) one another with respect to the middle points of all their transversal arguments.(Veronese) [2]

6. Two parallel straight lines intercept, on every transversal which passes trough the middle point of a segment included between them, another segment the middle point of which is the middle of the first (Ingami) [2]

7. Two straight lines that intersect one another cannot be parallel to a third line. (no 7 at [3] )

8. If two lines are parallel , then alternate internal angles cut by an transversal are congruent (converse alternate internal angle theorem). [4,6]

9. If t is a transversal to $l$ and $ l \parallel m $ and $ t \bot l $ then $t \bot m $. [4,6]

10. if $k \parallel l$ , $m \bot k$ and $n\bot l$, then either $m=n$ or $ m \parallel n$. [4]

11. Any two parallel lines have two common perpendicular lines. [5]

12. Any three distinct lines have a common transversal. [5]

13. There are not three lines such that any two of them are in the same side of the third. [5]

14. Two any parallel lines have a common perpendicular. [5]

15. Given $r,s$ lines, if $r$ is parallel to $s$, then $r$ is equidistant from $s$.[5]

16. Given a line $r$, the set of the points that are on the same side of $r$ and that are equidistant from $r$, is a line. [5]

17. Given lines $r,s,u,v$, if $r$ is parallel to $s$, $u$ is perpendicular to $r$ and $v$ is perpendicular to $s$, then $u$ and $v$ are parallel. [5,6]

18. Given lines $r,s,u,v$, if $r \perp s$, $s \perp u$ and $u \perp v$, then $r$ cuts $s$ (Bachmann Lottschnitt axiom). [5,6]

19. If $\overleftrightarrow{AB} \parallel \overleftrightarrow{CD}$ and $\overleftrightarrow{BC}$ is transversal to both of them such that $A$ and $D$ are in the same side of $\overleftrightarrow{BC}$, then $m(\measuredangle ABC) + m(\measuredangle DCB) = 180°$. [5]

20. For any point P, line l, with P not incident with l, and any line g, there exists a point G on g for which the distance to P exceeds the distance to l [8]

Triangles:

1. The sum of the angles in every triangle is 180° (triangle postulate).[1,6]

2. There exists a triangle whose angles add up to 180°.[1,6]

3. The sum of the angles is the same for every triangle.[1]

4. There exists a pair of similar, but not congruent, triangles.[1,6]

5. Every triangle can be circumscribed.[1,6]

6. In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides (Pythagoras' Theorem).[1]

7. There is no upper limit to the area of a triangle. (Wallis axiom)[1]

8. Given a triangle $\Delta ABC$, if $(AC)^2 = (AB)^2 + (BC)^2$, then $\angle B$ is a right angle. (converse of Pythagorean Theorem) [5]

9. Given a triangle $\Delta ABC$, exists $\Delta DEF$ such that $A \in \overline{DE}$, $B \in \overline{EF}$ and $C \in \overline{FD}$. [5]

10. Given a triangle $\Delta ABC$, if $D$ and $E$ are respectively the middle points of $\overline{AB}$ and $\overline{AC}$, then $DE = \frac{1}{2}BC$. [5]

11. (Thales) Given a triangle $\Delta ABC$, with $B$ in the circle of diameter $\overline{AC}$, then $\angle ABC$ is a right angle. [5,6]

12. The perpendicular bisectors of the sides of a triangle are concurrent lines. [5,6]

Rectangles:

1. There exists a quadrilateral such that the sum of its angles is 360°. (answer Ivo Terek below)

2. If three angles of a quadrilateral are right angles, then the fourth angle is also a right angle.[1,6]

3. There exists a quadrilateral in which all angles are right angles.[1,6]

4. The summit angles of the Saccheri quadrilateral are 90°. [1,6]

5. If in a quadrilateral 3 angles are right angles, the fourth is a right angle also.[2,6]

Circles:

1. Given any three points not on a straight line, there exists a circle trough them. (Legendre, Bolay)[2,6]

2. A curve of constant non-zero curvature is a circle.

3. A curve of constant non-zero curvature has finite extent.

4. There exist circles of arbitrarily low curvature.

5. The area of a circle grows at most polynomially in its radius.

Other:

1. Through any point within an angle less than 60° a straight line can always be drawn which meet both sides of the angle. (Legendre)[2]

2. Given an angle $\angle ABC$ and $D$ in its interior, every line that passes throuh $D$ cuts $\overrightarrow{BA}$ or $\overrightarrow{BC}$. [5,6]

3. If $A,B$ and $C$ are points of a circle with center $D$ such that $B$ and $D$ are in the same side of $\overleftrightarrow{AC}$, then $m(\measuredangle ABC) = \frac{1}{2}m(\measuredangle ADC)$. [5]

4. Given a acute angle $\angle ABC$ and $D \in \overrightarrow{BA}$, $D \neq B$, if $t$ contains $D$ and is perpendicular to $\overleftrightarrow{AB}$, then $t$ cuts $\overrightarrow{BC}$. [5]

References:

[1]: wikipedia http://en.wikipedia.org/wiki/Parallel_postulate

[2]: Heath's "Euclid, The Thirteen Books of The Elements" Dover edition

[3]: cut the knot http://www.cut-the-knot.org/triangle/pythpar/Fifth.shtml

[4]: Greenberg's "Euclidean and Non-Euclidean geometries" 3rd edition 1994

[5]: Professor Sergio Alves' notes of Non-Euclidean Geometry, from University of São Paulo (the original notes (in portuguese) in three images: here, here and here)

[6]: The computer checked proofs of the equivalence between 34 statements: http://geocoq.github.io/GeoCoq/html/GeoCoq.Meta_theory.Parallel_postulates.Euclid_def.html and the paper : https://hal.inria.fr/hal-01178236v2

[7]: Martin, The foundations of geometry and the non euclidean plane.

[8]: Pambuccian, Another equivalent of the Lotschnittaxiom, V. Beitr Algebra Geom (2017) 58: 167. doi:10.1007/s13366-016-0307-5

Answer 2

A few more:

• The sum of the angles in every quadrilateral is $360^\circ $.

• Exists a quadrilateral such that the sum of its angles is $360^\circ $.

• If two parallel lines are cut by a transversal line, then the alternate angles are congruent.

• Given lines $r,s,t$, if $r$ is parallel to $s$ and $t$ cuts $r$, then $t$ cuts $s$.

• Given lines $r,s,t$, if $r$ is parallel to $s$ and $s$ is perpendicular to $t$, then $t$ is perpendicular to $s$.

• Given lines $r,s,u,v$, if $r$ is parallel to $s$, $u$ is perpendicular to $r$ and $v$ is perpendicular to $s$, then $u$ and $v$ are parallel.

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Your $13$ is directly equivalent to saying that retangles exist. Remember that a Saccheri quadrilateral ${\bf S}ABCD$ is such that $\overline{AB} \cong \overline{CD}$ and $m(\measuredangle A) = m (\measuredangle D) = 90^\circ$. It can be proved also that $\overline{AD} \parallel \overline{BC}$ and $\measuredangle B \cong \measuredangle C$, among other stuff. A Lambert quadrilateral is one that have three right angles.

I suggest that you look around for neutral geometry, which consists os results that independ of the fifth postulate. A bit of hyperbolic geometry might be useful too, so you can detect where the uniqueness of the parallel was being used.

About the equivalences that use the notion of angle, in neutral geometry they are expressed using the notion of the defect of a triangle or a convex quadrilateral (I believe "defect" is the term used, since english is not my native language and I haven't seen any material about it in english) The defect of $\Delta ABC$ and $ABCD$ are, by definition: $$\delta (\Delta ABC) = 180 - (m(\measuredangle A) + m (\measuredangle B) + m (\measuredangle C))$$ and $$ \delta ( ABCD) = 180 - (m(\measuredangle A) + m (\measuredangle B) + m (\measuredangle C) + m (\measuredangle D))$$

Defect has a few additive properties, and sorta plays the role of "area" in non-Euclidean geometry. So in this fashion, you would have:

• 2. For every triangle $\Delta$, we have $\delta (\Delta) = 0 $.

• 3. There exists a triangle $\Delta$ such that $\delta(\Delta) = 0$.

• 4. Given any two triangles $\Delta_1$ and $\Delta_2$, we have $\delta (\Delta_1) = \delta (\Delta_2)$.

I will come back and add more equivalences here hopefully soon.

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EDIT.: More equivalences:

• If $\overleftrightarrow{AB} // \overleftrightarrow{CD}$ and $\overleftrightarrow{BC}$ is transversal to both of them such that $A$ and $D$ are in the same side of $\overleftrightarrow{BC}$, then $m(\measuredangle ABC) + m(\measuredangle DCB) = 180º$.

• If $\overleftrightarrow{AB} // \overleftrightarrow{CD}$ and $\overleftrightarrow{BC}$ is transversal to both of them such that $A$ and $D$ are in opposite sides of $\overleftrightarrow{BC}$, then $\measuredangle ABC \cong \measuredangle DCB$.

• The perpendicular bisectors of the sides of a triangle are concurrent lines.

• Given three non-collinear points, exists a circle which contains them.

• Exists a unique point equidistant from three non-collinear given points.

• Given a acute angle $\angle ABC$ and $D \in \overrightarrow{BA}$, $D \neq B$, if $t$ contains $D$ and is perpendicular to $\overleftrightarrow{AB}$, then $t$ cuts $\overrightarrow{BC}$.

• (Legendre) For every acute angle $\angle ABC$, and $D \in \mathrm{int}(\angle ABC)$, exists a line passing through $D$ that cuts both $\overrightarrow{BA}$ and $\overrightarrow{BC}$ in points distinct from $B$.

• (Thales) Given a triangle $\Delta ABC$, with $B$ in the circle of diameter $\overline{AC}$, then $\angle ABC$ is a right angle.

• If $A,B$ and $C$ are points of a circle with center $D$ such that $B$ and $D$ are in the same side of $\overleftrightarrow{AC}$, then $m(\measuredangle ABC) = \frac{1}{2}m(\measuredangle ADC)$.

• If $\angle ABC$ is a right angle, then $B$ is in the circle of diameter $\overline{AC}$.

• The perpendicular bisectors of the catheti of a right triangle are concurrent lines.

• Given lines $r,s,u,v$, if $r \perp s$, $s \perp u$ and $u \perp v$, then $r$ cuts $s$.

• Exists an acute angle $\angle ABC$ such that every line that contains $D \in \overrightarrow{BA}$, $D \neq B$, and it is perpendicular to $\overleftrightarrow{AB}$ also cuts $\overrightarrow{BC}$.

• Exists an acute angle such that for every point in its interior, passes a line through it that cuts both sides of the angle in points distinct from its vertex.

• Given $r,s$ lines, if $r$ is parallel to $s$, then $r$ is equidistant from $s$.

• Given a line $r$, the set of the points that are on the same side of $r$ and that are equidistant from $r$, is a line.

• Exists two distinct lines that are equidistant.

• (Wallis) Given a triangle $\Delta ABC$ and a line segment $DE$, exists a point $F$, with $D,E$ and $F$ non-collinear such that $\Delta DEF \cong \Delta ABC$.

• Exists two triangles that are similar, but not congruent.

• There is no absolute measure system.

• Any two parallel lines have two common perpendicular lines.

• The diagonals of a Saccheri quadrilateral intersect in their midpoints.

• Any three distinct lines have a common transversal.

• There are not three lines such that any two of them are in the same side of the third.

• Given a triangle $\Delta ABC$, if $D$ and $E$ are respectively the middle points of $\overline{AB}$ and $\overline{AC}$, then $DE = \frac{1}{2}BC$.

• Two any parallel lines have a common perpendicular.

• Given a triangle $\Delta ABC$, exists $\Delta DEF$ such that $A \in \overline{DE}$, $B \in \overline{EF}$ and $C \in \overline{FD}$.

• Given an angle $\angle ABC$ and $D$ in its interior, every line that passes throuh $D$ cuts $\overrightarrow{BA}$ or $\overrightarrow{BC}$.

• The Pythagorean Theorem.

• Given a triangle $\Delta ABC$, if $(AC)² = (AB)² + (BC)²$, then $\angle B$ is a right angle.

• Exists a triangle such its area is greater then any value given.

That's all I've got now.

EDIT: The original reference (in portuguese) in three images: here, here and here.

Answer 3

The notion of equivalence is relative to the theory and to the logic. It is important to note that some of the equivalences cited above are not correct if you do not assume Archimedes axiom or classical logic.

For example, in Dehn's semi-euclidean plane, the sum of angles of any triangle is $\pi$ but the axiom of parallels fails: https://en.wikipedia.org/wiki/Dehn_plane

Szmielew proved (assuming continuity) that every statement which is false in hyperbolic geometry and correct in Euclidean geometry is equivalent to the fifth parallel postulate

Wanda Szmielew. Elementary hyperbolic geometry. In P. Suppes L. Henkin and A. Tarski, editors, The axiomatic Method, with special reference to Geometry and Physics, pages 30–52, Amsterdam, 1959. North-Holland.

Let's call an Hilbert plane any model of either Group I II III of Hilbert's axioms or Tarski's axioms A1-A9.

The 34 statements listed bellow are all equivalent in any Archimedean Hilbert plane assuming classical logic. The equivalence can also be proved using a weaker assumption than Archimedes' axiom: Aristotle's axiom (see Old and New Results in the Foundations of Elementary Plane Euclidean and Non-Euclidean Geometries by Marvin Jay Greenberg).

Aristotle's axiom: Given any acute angle, any side of that angle, and any challenge segment PQ, there exists a point Y on the given side of the angle such that if X is the foot of the perpendicular from Y to the other side of the angle, then Y X > PQ.

If one drops Archimedes' axiom, then you obtain three separate groups of axioms:

1. Statements 1,2,5-20 are equivalent in any Hilbert plane
2. Statements 3,21-30 are equivalent in any Hilbert plane
3. Statements 4,31-33 are equivalent in any Hilbert plane

If one drops classical logic and only assume that for all points $A$ and $B$, $A=B \lor A \neq B$ then the first group is split into two:

1. Statements 1,13-20 are equivalent in any Hilbert plane without assuming classical logic.
2. Statements 2,5-13 are equivalent in any Hilbert plane without assuming classical logic.

List of statements:

1. (Tarski's parallel postulate) Given a point $D$ between the points $B$ and $C$ and a point $T$ further away from $A$ than $D$ on the half line $AD$, one can build a line which goes through $T$ and intersects the sides $BA$ and $BC$ of the angle $\angle ABC$ respectively further away from $B$ than $A$ and $C$.

2. (Playfair's postulate) There is a unique parallel to a given line going through some point.

3. (Triangle postulate) The sum of the angles of any triangle is two right angles.

4. (Bachmann's Lotschnittaxiom) Given the lines $l$, $m$, $r$ and $s$, if $l$ and $r$ are perpendicular, $r$ and $s$ are perpendicular and $s$ and $m$ are perpendicular, then $l$ and $m$ must meet.

5. (Postulate of transitivity of parallelism) If two lines are parallel to the same line then these lines are also parallel.

6. (Midpoint converse postulate) The parallel line to one side of a triangle going through the midpoint of another side cut the third side in its midpoint.

7. (Alternate interior angles postulate) The line falling on parallel lines makes the alternate angles equal one to one another.

8. (Consecutive interior angles postulate) A line falling on parallel lines makes the sum of interior angles on the same side equal to two right angles.

9. (Perpendicular transversal postulate) Given two parallel lines, any line perpendicular to the first line is perpendicular to the second line.

10. (Postulate of parallelism of perpendicular transversals) Two lines, each perpendicular to one of a pair of parallel lines, are parallel.

11. (Universal Posidonius' postulate) If two lines are parallel then they are everywhere equidistant.

12. (Alternative Playfair's postulate) Any line parallel to line $l$ passing through a point $P$ is equal to the line that passes through $P$ and shares a common perpendicular with $l$ going through $P$.

13. (Proclus' postulate) If a line intersects one of two parallel lines then it intersects the other.

14. (Alternative Proclus' postulate) If a line intersects in $P$ one of two parallel lines which share a common perpendicular going through $P$, then it intersects the other.

15. (Triangle circumscription principle) For any three non-collinear points there exists a point equidistant from them.

16. (Inverse projection postulate) For any given acute angle, any point together with its orthogonal projection on one side of the angle form a line which intersects the other side.

17. (Euclid 5) Given a non-degenerated parallelogram $PRQS$ and a point $U$ strictly inside the angle $\angle QPR$, there exists a point $I$ such that $Q$ and $U$ are respectively strictly between $S$ and $I$ and strictly between $P$ and $I$.

18. (Strong parallel postulate) Given a non-degenerated parallelogram $PRQS$ and a point $U$ not on line $PR$, the lines $PU$ and $QS$ intersect.

19. (Alternative strong parallel postulate) If a straight line falling on two straight lines make the sum of the interior angles on the same side different from two right angles, the two straight lines meet if produced indefinitely.

20. (Euclid's parallel postulate) If a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

21. (Postulate of existence of a triangle whose angles sum to two rights) There exists a triangle whose angles sum to two rights.

22. (Posidonius' postulate) There exists two lines which are everywhere equidistant.

23. (Postulate of existence of similar but non-congruent triangles) There exists two similar but non-congruent triangles.

24. (Thales' postulate) If the circumcenter of a triangle is the midpoint of a side of a triangle, then the triangle is right.

25. (Thales' converse postulate) In a right triangle, the midpoint of the hypotenuse is the circumcenter.

26. (Existential Thales' postulate) There is a right triangle whose circumcenter is the midpoint of the hypotenuse.

27. (Postulate of right Saccheri quadrilaterals) The angles of any Saccheri quadrilateral are right.

28. (Postulate of existence of a right Saccheri quadrilateral) There is a Saccheri quadrilateral whose angles are right.

29. (Postulate of right Lambert quadrilaterals) The angles of any Lambert quadrilateral are right i.e. if in a quadrilateral three angles are right, so is the fourth.

30. (Postulate of existence of a right Lambert quadrilateral) There exists a Lambert quadrilateral whose angles are all right.

31. (Weak inverse projection postulate) For any angle, that, together with itself, make a right angle, any point together with its orthogonal projection on one side of the angle form a line which intersects the other side.

32. (Weak Tarski's parallel postulate) For every right angle and every point $T$ in the interior of the angle, there is a point on each side of the angle such that $T$ is between these two points.

33. (Weak triangle circumscription principle) The perpendicular bisectors of the legs of a right triangle intersect.

34. (Legendre's parallel postulate) There exists an acute angle such that, for every point $T$ in the interior of the angle, there is a point on each side of the angle such that $T$ is between these two points.