Proving that three non-aligned points determine a unique plane

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I started to study the axioms of Euclidean Geometry and i wanted to prove by myself a theorem (using only the axioms and rules of Euclidean Geometry):




For 3 non-aligned points passes a unique plane.




However, I don't know how to do it. Can you help me?



ps: I found the previous statement (which i called "theorem) called "corollary". To me theorems and corollaries seems the same thing. Am I wrong ?




geometry proof-writing euclidean-geometry




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  • 1




    The classical 5 axioms of Euclidean Geometry are about planar geometry. You want to prove something about space geometry. Can you indicate what are the axioms you are working with?
    – A. Arredondo
    2 days ago






  • 1




    As for "theorems" vs "corollaries" ... Both are indeed "the same thing", in that they are statements that are proven (as opposed to axioms or postulates, which are assumed). A corollary is typically a statement that follows easily from some result that one has just proven; it's like an added bonus for the work that was done. The term "corollary" describes a relationship between results as they happen to be presented by a particular author. What one author calls a corollary, another might call simply a theorem, because that second author proved the result directly.
    – Blue
    2 days ago










  • So the statement i writed in main post is not a theorem but an axiom ? The italian wikipedia page about Euclidean Geometry calls it corollary
    – Koinos
    2 days ago











  • @Koinos I take issue with the cited Italian Wikipedia article. The axioms do not mention the concept of a plane, so you can't really prove anything regarding a plane using the axioms.
    – A. Arredondo
    2 days ago














up vote
0
down vote

favorite












I started to study the axioms of Euclidean Geometry and i wanted to prove by myself a theorem (using only the axioms and rules of Euclidean Geometry):




For 3 non-aligned points passes a unique plane.




However, I don't know how to do it. Can you help me?



ps: I found the previous statement (which i called "theorem) called "corollary". To me theorems and corollaries seems the same thing. Am I wrong ?




geometry proof-writing euclidean-geometry




share|cite|improve this question

















  • 1




    The classical 5 axioms of Euclidean Geometry are about planar geometry. You want to prove something about space geometry. Can you indicate what are the axioms you are working with?
    – A. Arredondo
    2 days ago






  • 1




    As for "theorems" vs "corollaries" ... Both are indeed "the same thing", in that they are statements that are proven (as opposed to axioms or postulates, which are assumed). A corollary is typically a statement that follows easily from some result that one has just proven; it's like an added bonus for the work that was done. The term "corollary" describes a relationship between results as they happen to be presented by a particular author. What one author calls a corollary, another might call simply a theorem, because that second author proved the result directly.
    – Blue
    2 days ago










  • So the statement i writed in main post is not a theorem but an axiom ? The italian wikipedia page about Euclidean Geometry calls it corollary
    – Koinos
    2 days ago











  • @Koinos I take issue with the cited Italian Wikipedia article. The axioms do not mention the concept of a plane, so you can't really prove anything regarding a plane using the axioms.
    – A. Arredondo
    2 days ago












up vote
0
down vote

favorite









up vote
0
down vote

favorite











I started to study the axioms of Euclidean Geometry and i wanted to prove by myself a theorem (using only the axioms and rules of Euclidean Geometry):




For 3 non-aligned points passes a unique plane.




However, I don't know how to do it. Can you help me?



ps: I found the previous statement (which i called "theorem) called "corollary". To me theorems and corollaries seems the same thing. Am I wrong ?




geometry proof-writing euclidean-geometry




share|cite|improve this question













I started to study the axioms of Euclidean Geometry and i wanted to prove by myself a theorem (using only the axioms and rules of Euclidean Geometry):




For 3 non-aligned points passes a unique plane.




However, I don't know how to do it. Can you help me?



ps: I found the previous statement (which i called "theorem) called "corollary". To me theorems and corollaries seems the same thing. Am I wrong ?





geometry proof-writing euclidean-geometry





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited 2 days ago









Blue

43.6k868141




43.6k868141









asked 2 days ago









Koinos

485




485







  • 1




    The classical 5 axioms of Euclidean Geometry are about planar geometry. You want to prove something about space geometry. Can you indicate what are the axioms you are working with?
    – A. Arredondo
    2 days ago






  • 1




    As for "theorems" vs "corollaries" ... Both are indeed "the same thing", in that they are statements that are proven (as opposed to axioms or postulates, which are assumed). A corollary is typically a statement that follows easily from some result that one has just proven; it's like an added bonus for the work that was done. The term "corollary" describes a relationship between results as they happen to be presented by a particular author. What one author calls a corollary, another might call simply a theorem, because that second author proved the result directly.
    – Blue
    2 days ago










  • So the statement i writed in main post is not a theorem but an axiom ? The italian wikipedia page about Euclidean Geometry calls it corollary
    – Koinos
    2 days ago











  • @Koinos I take issue with the cited Italian Wikipedia article. The axioms do not mention the concept of a plane, so you can't really prove anything regarding a plane using the axioms.
    – A. Arredondo
    2 days ago












  • 1




    The classical 5 axioms of Euclidean Geometry are about planar geometry. You want to prove something about space geometry. Can you indicate what are the axioms you are working with?
    – A. Arredondo
    2 days ago






  • 1




    As for "theorems" vs "corollaries" ... Both are indeed "the same thing", in that they are statements that are proven (as opposed to axioms or postulates, which are assumed). A corollary is typically a statement that follows easily from some result that one has just proven; it's like an added bonus for the work that was done. The term "corollary" describes a relationship between results as they happen to be presented by a particular author. What one author calls a corollary, another might call simply a theorem, because that second author proved the result directly.
    – Blue
    2 days ago










  • So the statement i writed in main post is not a theorem but an axiom ? The italian wikipedia page about Euclidean Geometry calls it corollary
    – Koinos
    2 days ago











  • @Koinos I take issue with the cited Italian Wikipedia article. The axioms do not mention the concept of a plane, so you can't really prove anything regarding a plane using the axioms.
    – A. Arredondo
    2 days ago







1




1




The classical 5 axioms of Euclidean Geometry are about planar geometry. You want to prove something about space geometry. Can you indicate what are the axioms you are working with?
– A. Arredondo
2 days ago




The classical 5 axioms of Euclidean Geometry are about planar geometry. You want to prove something about space geometry. Can you indicate what are the axioms you are working with?
– A. Arredondo
2 days ago




1




1




As for "theorems" vs "corollaries" ... Both are indeed "the same thing", in that they are statements that are proven (as opposed to axioms or postulates, which are assumed). A corollary is typically a statement that follows easily from some result that one has just proven; it's like an added bonus for the work that was done. The term "corollary" describes a relationship between results as they happen to be presented by a particular author. What one author calls a corollary, another might call simply a theorem, because that second author proved the result directly.
– Blue
2 days ago




As for "theorems" vs "corollaries" ... Both are indeed "the same thing", in that they are statements that are proven (as opposed to axioms or postulates, which are assumed). A corollary is typically a statement that follows easily from some result that one has just proven; it's like an added bonus for the work that was done. The term "corollary" describes a relationship between results as they happen to be presented by a particular author. What one author calls a corollary, another might call simply a theorem, because that second author proved the result directly.
– Blue
2 days ago












So the statement i writed in main post is not a theorem but an axiom ? The italian wikipedia page about Euclidean Geometry calls it corollary
– Koinos
2 days ago





So the statement i writed in main post is not a theorem but an axiom ? The italian wikipedia page about Euclidean Geometry calls it corollary
– Koinos
2 days ago













@Koinos I take issue with the cited Italian Wikipedia article. The axioms do not mention the concept of a plane, so you can't really prove anything regarding a plane using the axioms.
– A. Arredondo
2 days ago




@Koinos I take issue with the cited Italian Wikipedia article. The axioms do not mention the concept of a plane, so you can't really prove anything regarding a plane using the axioms.
– A. Arredondo
2 days ago















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