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Volume One: Geometry without Multiplication

Victor Aguilar

posted on 20 January 2016

download (3656 views, 461 download, 9 comments)

I propose to write a textbook on the subject of Euclidean geometry.  It will be divided into two volumes, Geometry without Multiplication and Geometry with Multiplication.


I invite comment from other geometers on the foundational material, definitions, proofs that I have done so far and on the organization of the theorems.


Are there any theorems missing?  Are there any out of logical sequence?


At the end I conclude with some theorems of practical use that can be proven without recourse to multiplication.  Are there other practical uses of this theory that you can suggest?


This is an ongoing project and I've already made a change, so please download it again.


I moved SSS to the beginning and proved SAS using it, rather than the other way around as Euclid did.  I think a lot of people are uncomfortable with the use of superposition to prove SAS at the very beginning since sliding figures around and on top of each other was not in Euclid's foundation.  But I think SSS can be proven based just on my foundation and then SAS is an easy corollary of it.


In a week or so I will post the actual book here at the Econophysics Forum and then update it every week or two as I write it.  If you have any comments or suggestions, please let me know.


"It is a completely mistaken idea that scientific theory is based on deductions from a series of postulates – that is the description of the methodology of mathematics…  There is no science which uses axioms and logical deductions to derive scientific theory."


The World Economics Association journal editor who wrote this is feeding off hatred leaned in high school.  Where else?  Neither he nor his legion of followers have ever taken a proof-oriented mathematics class at a university.  They could not have learned hate for deductive logic anywhere else but in high school.  If this were just one crazy guy, it would not matter.  But the fact that he holds a powerful editorship and his followers encompass close to 99% of high school graduates implies that there are systemic problems with how deductive logic is presented to high school students.


It is my goal to correct these problem.

I have now changed it again to give high school geometers ranks:


White Belt:                  Abstract Algebra

Orange Belt:               Triangle Congruence 

Green Belt:                  Quadrilaterals

Red Belt:                     Quadrature

Black Belt:                   Circles


I have written the text for white- and orange-belt students and I have written an outline of the remaining textbook.  I will update as I progress.  I hope to receive comments on my proofs and on my organization of theorems from other geometers during the writing of this book.


Why the martial arts analogy?  We are under attack from the math haters.  Asad Zaman writes:


It is a completely mistaken idea that scientific theory is based on deductions from a series of postulates – that is the description of the methodology of mathematics…  There is no science which uses axioms and logical deductions to derive scientific theory.


“Mathematics is not a ‘science’ since it is not based in any direct way on observational evidence.  Unlike scientific laws, mathematical laws are not affirmed observational evidence.  Recognition of the possibility that there are bodies of knowledge which are not science would lead to greater tolerance and pluralism which is currently desperately needed.”


Tolerance?  No.  Such outrageous talk by the World Economics Association has brought war and discord to our once quiet study of triangles.  If Asad Zaman were just some crazy man, it would not matter.  But Zaman is the editor of the Real World Economics Review.  He wields this terrifying power to no other end than to ban all mention of deductive logic.  I alone stand in his path.  If I falter, then geometry will be excluded from the high school curriculum and a whole generation will be taught only hate for deductive logic.


Can you help me fight the Islamic extremists?  Intimidating their enemies by cutting off heads and baffling them by slyly redefining dialectical materialism as “ontology” and historical materialism as “processual,” the Islamists and the Marxists have united to rid the world of geometry and to raise a generation of Westerners with only hate for deductive logic.


Logic alone can defeat the haters!

Green belt theorems!

Red belt theorems!

Blue belt theorems!

Yellow belt theorems!


The organization is now as follows:




White Belt:       Foundations


Orange Belt:    Congruence


Yellow Belt:      Parallelism I




 Green Belt:       Parallelism II


 Red Belt:          Quadrature


 Blue Belt:          Tangents    




Cho–Dan:           Similarity


 Problem Solving


 Application:  Optics


The book is written through blue belt.  The black belt chapter is outlined.  There is a list of all the theorems proven so far on page 82.

Geometry-Do update


I have added some theorems to the red-belt chapter  and switched its position with the blue-belt chapter.  Also, I have added quite a few new problems and constructions that are intended to help students training for the International Mathematical Olympiad.

1st year geometry is done, I believe.  This is white-, yellow- and orange-belt.  There is a 20-question green-belt entrance exam on page 82.  I would very much appreciate if people here at the Econophysics form could spend a few minutes taking this exam and report to me if it reflects what is expected of high school geometry students in your country after one year of study.

Geometry-Do almost done.


Today's update almost completes Volume One.  I expect Volume One:  Geometry without Multiplication to be finished in a few weeks.  It will be published as a paperback and later - probably next year - I will publish a hardback that includes the blue and black belt chapters, which introduce multiplication.


I have gained a bit of a following at Research Gate, but nobody from the Econophysics Forum has commented.  If there is anybody here with an interest in geometry, I would very much appreciate your comments!