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An Alternative Postulate Set for Geometry

Victor Aguilar

posted on 15 June 2016

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The purpose of this paper is to introduce a new set of postulates for geometry and to define the minimum of abstract algebra axioms needed. Emphasis is put on accepting as intuitive only those spatial relations that small children understand without explanation; their parents are just assigning names to concepts that are instinctive in humans. Geometers are invited to compare these assumptions with the foundations that are used in other textbooks. These postulates and axioms are to be the foundations of a textbook, Geometry–Do.

Discussion

Euclid's axioms are not enough and I have added two more. My axioms are now as follows:

1. Two points fully define a segment.
2. Three points fully define a triangle.
3. A segment fully defines a line.
4. A triangle’s area is zero if and only if its vertices are collinear.
5. The center and the radius fully define a circle.
6. All right angles are equal to each other.
7. A line and a point not on it fully define the parallel through that point.

I uploaded a new version.

 

My postulates are now six:

 

Segment             Two points fully define a segment.
Triangle              Three points fully define a triangle.
Line                     A segment fully defines a line.
Circle                  The center and the radius fully define a circle.
Right Angle        All right angles are equal to each other.
Parallel                A line and a point not on it fully define the parallel through that point.

 

Also, the new version has a list of theorem through red belt.

This will soon be published in an education journal.  It has been rewritten extensively since it was first posted here.  Any comments that you have before the paper goes to print would be very much appreciated.