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I'm new here and this is my first post.

I'm not sure whether this thread is where it should be, but the post doesn't deal with homework...I'm working on my own at the moment, and the things I'd like to present here is really a private business, so I decided to post them in the general maths section...I hope they will be of interest to you.

I made up my mind to revise the whole course of Euclid geometry once more...but I'm doing it from a slightly different perspective. The problems I'd like to offer are actually simple...but I solve them using pure axiomatic method, so I pay attention to every detail...i.e. even if something is obvious and clear I try to prove it (except axioms).

Please, check the following solutions for a number of problems and say whether they are correct...or, in addition, whether any other solution is possible.

I'm dealing with solid geometry at the moment...so the whole thing is based on the three main axioms of geometry in 3-dimensional space, which come in addition to those of plane geometry:

Axiom 1. There exist points, which belong to a plane and which do not;

Axiom 2. If two different planes have a common point, then they intersect each other at the line, containing this point.

Axiom 3. If two different lines have a common point, then they set a unique plane, which contains the two lines.

1. Problem:

The points A, B, C all lie in each of two different planes (e.g. alpha and betta). Prove that these points lie on the same line (e.g. "a").

Proof:

(Axiom 1 is subtely involved in each sentence and in what is given, since it allows us to operate with points and planes, so there seems to be no need to mention it anymore)

1) According to Axiom 2, if two different planes have a common point, then they intersect each other at the line, containing this point. It follows that if the points A, B, C lie in the plane alpha then they also lie in some planes, which intersect alpha at the lines, containing the points A, B, C.

2) It is given that the points also lie in betta, then the three planes, containing the points A, B, C are actually the same plane betta (I imagine them merging together to form the plane betta). Hence, the three lines, which are set by the three planes are the intersection line "a" of alpha and betta (the lines merge to the unique line). It finally follows that all three points A, B, C belong to the same line "a".

q.e.d.

I think a proof by contradiction is also possible here.

Suppose the points A, B, C do not lie on the same line a (Axiom 1 of plane geometry), then, since they lie in alpha (this is given), they also lie in some planes, which intersect alpha at some lines, containing these points (Axiom 2), which is impossible, because according to what is given all three points A, B and C must also lie in the plane betta, diffrent from alpha and intersecting it (this is given).

It's pure logic actually. I hope the proofs are consistent. I understand that this all might seem nonsense to you, but I just want to learn how to work with detailed proofs and weird mathematical structures.

If you have an interest in such things there is more to come.

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# Some solid geometry

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