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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

  1. $|ab|=|cd|$.
  2. Lines $ab$ and $cd$ are parallel.

  3. 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 $ae$.

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  • $\begingroup$ At this point of the theory, do you have the notion of a line segment? If so, you may want to show (and use) the fact that the line segments $[ac]$ and $[ce]$ lie on the same side of the line $ae$. $\endgroup$ – paf Aug 25 '16 at 22:12
  • $\begingroup$ Yes, I have this notion and I know how to show this fact. But I still don't know how to prove the thesis. $\endgroup$ – Kulisty Aug 26 '16 at 8:21
  • $\begingroup$ Maybe the following 3-step-approach will lead to success: Step 1: Define a relation $\sim$ by $(a,b)\sim(c,d):\Leftrightarrow\exists v:a+v=c\ \land\ b+v=d $! Step 2: Prove that $\sim$ is an equivalence relation! (This is rather easy.) Step 3: Prove that the definition of $\sim$ is equivalent to your original definition! $\endgroup$ – Xaver Aug 27 '16 at 15:18
  • $\begingroup$ Yes, but how to define this addition? I can't add points. And I think there should be $a+v=b \wedge c+v=d$. $\endgroup$ – Kulisty Aug 28 '16 at 7:47
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What you want to prove using Hilbert's axioms has been proved by Gabriel Braun using Tarski's axioms within GeoCoq (the proofs have been checked by a machine).

The definitions of equality of vectors which seems to be equivalent to yours, can be found here: https://github.com/GeoCoq/GeoCoq/blob/master/Tarski_dev/Definitions.v In Coq's syntax: Definition EqV A B C D := Parallelogram A B D C / A = B /\ C = D.

Then you can find the proofs that it is an equivalence relation here: https://github.com/GeoCoq/GeoCoq/blob/master/Tarski_dev/Annexes/vectors.v It relies on a property of parallelograms that Gabriel Braun has proved in Coq and described in french here: http://gabrielbraun.free.fr/Geometry/Tarski/plg_pseudo_trans.html#/start

Tarski's axioms and Hilbert's axiom (excluding continuity axioms) are mutually interpretable, so the same proof can be done using Hilbert's axioms.

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