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]
There is at most one line that can be drawn parallel to another given one through an external point. (Playfair's axiom)[1,6]
There exists a pair of straight lines that are at constant distance from each other.[1]
Two lines that are parallel to the same line are also parallel to each other.[1,6]
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]
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]
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]
Two straight lines that intersect one another cannot be parallel to a third line. (no 7 at [3] )
If two lines are parallel , then alternate internal angles cut by an transversal are congruent (converse alternate internal angle theorem). [4,6]
If t is a transversal to $l$ and $ l \parallel m $ and $ t \bot l $ then $t \bot m $. [4,6]
if $ k \parallel l $ , $ m \bot k $ and $ m \bot l $ then either $ m=n $ or $ m \parallel n.$ [4]
Any two parallel lines have two common perpendicular lines. [5]
Any three distinct lines have a common transversal. [5]
There are not three lines such that any two of them are in the same side of the third. [5]
Two any parallel lines have a common perpendicular. [5]
Given $r,s$ lines, if $r$ is parallel to $s$, then $r$ is equidistant from $s$.[5]
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]
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]
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]
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]
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:
The sum of the angles in every triangle is 180° (triangle postulate).[1,6]
There exists a triangle whose angles add up to 180°.[1,6]
The sum of the angles is the same for every triangle.[1]
There exists a pair of similar, but not congruent, triangles.[1,6]
Every triangle can be circumscribed.[1,6]
In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides (Pythagoras' Theorem).[1]
There is no upper limit to the area of a triangle. (Wallis axiom)[1]
Given a triangle $\Delta ABC$, if $(AC)^2 = (AB)^2 + (BC)^2$, then $\angle B$ is a right angle. (converse of Pythagorean Theorem) [5]
Given a triangle $\Delta ABC$, exists $\Delta DEF$ such that $A \in \overline{DE}$, $B \in \overline{EF}$ and $C \in \overline{FD}$. [5]
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]
(Thales) Given a triangle $\Delta ABC$, with $B$ in the circle of diameter $\overline{AC}$, then $\angle ABC$ is a right angle. [5,6]
The perpendicular bisectors of the sides of a triangle are concurrent lines. [5,6]
Rectangles:
There exists a quadrilateral such that the sum of its angles is 360°. (answer Ivo Terek below)
If three angles of a quadrilateral are right angles, then the fourth angle is also a right angle.[1,6]
There exists a quadrilateral in which all angles are right angles.[1,6]
The summit angles of the Saccheri quadrilateral are 90°. [1,6]
If in a quadrilateral 3 angles are right angles, the fourth is a right angle also.[2,6]
Circles:
Given any three points not on a straight line, there exists a circle trough them. (Legendre, Bolay)[2,6]
A curve of constant non-zero curvature is a circle.
A curve of constant non-zero curvature has finite extent.
There exist circles of arbitrarily low curvature.
The area of a circle grows at most polynomially in its radius.
Other:
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]
Given an angle $\angle ABC$ and $D$ in its interior, every line that passes throuh $D$ cuts $\overrightarrow{BA}$ or $\overrightarrow{BC}$. [5,6]
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]
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
There is some confusion about these questions. So I state here some facts:
- The axioms of Euclid (minus 5) prove that parallels exist.
- Playfair's postulate is that through a point not on a line there is ONLY ONE parallel. This is equivalent to the statement of Euclid's fifth postulate.
- $\Delta=\pi$ (that is the sum of the angles of a triange is $\pi$ radians) DOES NOT imply the parallel postulate without the Archimedean axiom.
- If one assumes that Archimedean axiom then $\Delta=\pi \Rightarrow $Playfair.
The proof of the last statement is as follows:
Let $A$ be a point not on line $l$, drop perpendicular from $A$ to $l$ at $B$.
Take the line $m$ through $A$ perpendicular to $AB$, this is parallel to $l$ as Euclid shows. If there were a second line through $A$ parallel to $l$, say making angle $\beta$ with $m$, then take a point $C$ on $l$ such that $\angle ACB <\beta$ (this is where the Archimedean axiom is used). Now the right angle triangle $ABC$ as angle sum $<\pi$ as one sees form the picture.
Best Answer
I know two ways of proving that something is not provable: - Using the semantic approach, one builds a counter model, i.e. in which all other axioms hold but the parallel postulate fails (this is the usual method for the parallel postulate). - Using the syntactic method, you can show that something is not provable more directly.
Together with Michael Beeson and Pierre Boutry, we have given a syntactic proof of the independence of the parallel postulate in the context of Tarski's axioms : Herbrand's theorem and non-Euclidean geometry, Bulletin of Symbolic Logic, Association for Symbolic Logic, 2015, 21 (2), pp.12
The general idea of the proof is that the parallel postulate (at least some versions of this postulate) allows to construct points which are arbitrarily far from the given points (take two lines which are very close to be parallel), whereas other axioms allows only to double the maximum distance between the points constructed so far.
To explain the difference between a syntactic independence proof and a semantic one, I give an elementary example in the following talk: http://dpt-info.u-strasbg.fr/~narboux/slides/Herbrand-Euclid-vulgarization.pdf