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I did not study the detail history of Euclid. As far as I know, the geometry starts from a few postulates and keeps on expanding (through theorems developing) to the present state. Even with these theorems as tools, proving some facts (in some geometrical problems) still is very clumsy. I am thinking “Is there anything we can include in it” to makes it ‘stronger’ so that proofs becomes easier.

Let me give a simple example (and I hope I am correct). In my school days, we were using Durell’s book. In which, the ideas of “transformations” were never taught. Thus, adding them to the Euclidean geometry makes it ‘stronger’.

Restrictions:- (1) by geometry, I mean plane geometry; (2) "Euclidean geometry" has been used as keyword to search through SE, and re-direction to this site may not be necessary.

Mick
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  • Could you precise the meaning of "clumsy"? You should provide also precisions on what kind of things should be added. Note that the existence of transformation is a consequence of the axioms of geometry. If the book looks clumsy without explicitely mentionning transformations, it is not the problem of the Euclidean geometry. – Taladris Aug 09 '14 at 04:46
  • Also, why do you want to restrict to plane geometry? Considering higher dimensions is not really a problem and usually provides richer problems (example: the combinatorics of polyhedrons/polytopes vs the combinatorics of polygons). – Taladris Aug 09 '14 at 04:50
  • @Taladris By "clumsy", I don't mean the book is clusmy. I just mean I need quite a bit of effort, to prove some simple facts, before I can ensure that others can understand what I mean. – Mick Aug 09 '14 at 04:51
  • @Taladris Regarding the restriction on plane geometry only, I think that it is a higher school (and examine-able) topic. Raising it to higher dimensions may not be suitable for the high school goers. – Mick Aug 09 '14 at 05:07

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First, the idea of "transformations" is not something that was added to Euclidean geometry; rather, the concept of Euclidean geometry as the study of invariants of Euclidean space under affine transformations is simply a more modern take on the five axioms.

There are certainly studies of the underlying axiomatic system of Euclidean geometry, in fact it is quite famously the oldest axiomatic system that endured scrutiny throughout mathematical history. Modern takes on this are a bit of an ongoing topic of research; Hilbert, Tarski, and Birkhoff each came up with their own axiomatic systems to describe Euclidean geometry. It would really fall under the category of logic/set theory rather than geometry though. If you are interested in what theorems of Euclidean geometry depend on which axioms, and how strengthening/weakening the axioms affects the theory, you should check out this line of thought.

For a possibly relevant (though not duplicate) question, check out Research in plane geometry or euclidean geometry.

Community
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Gyu Eun Lee
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  • (1) I don’t agree with your first paragraph. For example, in proving some problems, sometimes we will write “… by symmetry….” to shorten our proof. This is what I consider as added tool to that geometry. (2) Glad to hear that various researches are going on this topic. (3) However, my question has restrictions and re-writing another axiomatic approach is not what I am referring to. – Mick Aug 09 '14 at 05:16
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    I'm not sure what you mean by "added tool" then. Using the term "symmetry" in a plane geometry proof is shorthand for "this property is invariant under affine transformation," but an abbreviation is hardly an "added tool" mathematically speaking. Please clarify your question in the original post, as I think other users are also having trouble understanding your intent. – Gyu Eun Lee Aug 09 '14 at 05:24
  • @Mick: I thin E.G. was dead after introducing Non E.G versions. – Mikasa Aug 09 '14 at 06:12
  • @BabakS Sad to here that. – Mick Aug 10 '14 at 03:42
  • @kigen I use “Euclidthegame” as example. At L-1, player has just 4 basic tools. If the player can use them to construct the midpoint of … , then he is awarded with an extra tool - the midpoint tool, so that drawing all midpoints in the future need not be started from square-1. Likewise, (1) adjacent angle …, and (2) angle sum ... are tools added to the postulates. “Ext. angle of ⊿” is one on top of (1) & (2). “Power of a point” is another on top of all theorems before it. Because the idea of "symmetry" worked in EG, it forms another added tool. In short, will there be more tools? – Mick Aug 10 '14 at 05:08
  • @Mick All of these "tools" you refer to are theorems of plane geometry, deducible from the axioms. Are you asking whether there are more theorems to be proved in plane geometry? Because in that case, contrary to Babak's belief, classical Euclidean geometry is far from dead; there is even a journal dedicated to it, see http://forumgeom.fau.edu/. – Gyu Eun Lee Aug 10 '14 at 21:12
  • @kigen That is what I am mean - having more (but must be applicable) theorems developed making ... easier. – Mick Aug 11 '14 at 04:14