Axiom:Axiom of Triangle Existence

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Axiom

Let $\equiv$ be the relation of equidistance.

Let $\mathsf{B}$ be the relation of betweenness.


Then the following axiom holds:

$\forall a,a',b,b',c',p: ab \equiv a'b' \implies$
$\exists c,x: \left({ac \equiv a'c' \land bc \equiv b'c' \land \mathsf{B}cxp}\right) \land \left({\mathsf{B}abx \lor \mathsf{B}bxa \lor \mathsf{B}xab}\right)$

where $a, b, c, a', b', c', x, p$ are points.


Intuition

AxiomOfTriangleExistence.png

Draw two congruent line segments, $ab$, $a'b'$.

Let the three points $a',c',b'$ form a triangle.

Pick any point $p$.

Then it is always possible to extend $p$ into some line segment $pc$ passing through segment $ab$ at a point $x$.

Further, it is possible to pick a point $c$ such that a triangle $abc$ is formed congruent to triangle $a'b'c'$.


Note that this axiom still holds in degenerate cases.

For example, if $\triangle{abc} = \triangle{a'b'c'}$ (where $=$ denotes logical equality), then this axiom may be interpreted as "it is possible to drop a line segment from the apex of a triangle passing through the base of the triangle".

Sources