I'm working in Hilbert's axioms of Euclidean plane geometry. I have problems with proving one thing concerning vectors. My definition of the vector is as follows:
Vector $\overrightarrow{ab}$ is an ordered pair of points $(a,b)$.
Now I define a relation between vectors $\overrightarrow{ab}$ and $\overrightarrow{cd}$:
If $a=b$ and $c=d$, the vectors are in relation.
If $a\neq b$ and $c=d$, the vectors are not in relation.
If $a=b$ and $c\neq d$, the vectors are not in relation.
If $a\neq b$ and $c\neq d$, the vectors are in relation if and only if all following conditions are true
- $|ab|=|cd|$.
- Lines $ab$ and $cd$ are parallel.
a. If the lines $ab$ and $cd$ are equal, halfline $ab$ is contained in halfline $cd$ or halfline $cd$ is contained in halfline $ab$.
b. If the lines $ab$ and $cd$ are disjoint, points $b$ and $d$ lie on the same side of the line $ac$ (in other words in the same halfplane whose border is line $ac$)
Now I want to prove this is an equivalence relation (equivalence classes would be free vectors). The problem is with transitivity in the case when all lines are disjoint. I need to prove:
Let $ab$, $cd$, $ef$ be parallel lines. $b$ and $d$ lie on the same side of the line $ac$, $d$ and $f$ lie on the same side of the line $ce$. Then $b$ and $f$ lie on the same side of the line $ae$.