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Geometry–Do Volume One: Geometry without Multiplication by Victor Aguilar, author of Axiomatic Theory of Economics Chapters Postulates White Belt Foundations 1. Two points fully define the segment between them. Yellow Belt Congruence 2. By extending it, a segment fully defines a line. Orange Belt Parallelograms 3. Three noncollinear points fully define a triangle. Green Belt Triangle Construction 4. The center and the radius fully define a circle. Red Belt Triangles, Advanced 5. All right angles are equal; equivalently, all straight Blue Belt Quadrature Theory angles are equal. Cho–Dan Harmonic Division 6. A line and a point not on it fully define the parallel Yi–Dan Circle Inversion through that point. This book was NOT funded by the Bill and Melinda Gates Foundation. A journey of discovery? A death march? It’s the same thing! Copyright ©2020 This PDF file, which consists only of the foundational pages and the index, is the ONLY authorized free PDF file for Geometry–Do. Do not accept PDF files that do not have the copyright notice. www.researchgate.net/publication/291333791_Volume_One_Geometry_without_Multiplication Volume One, white through red belt, is available as a bound paperback, as is an abridgement consisting of white, yellow and early orange belt. Free review copies are available to mathematicians. Teachers can purchase individual copies from book sellers or contact me directly for quantity discounts, but I do not need the reviews of people with education degrees, so no free review copies are available to them. A math degree is a degree granted by a university mathematics department; people with an education degree (granted by a university education department) should not call it a math degree; doing so will not get them a free review copy. The order page and the contact page – for questions or if you wish to endorse the book – are at my website, shown below. Research Gate members may private message me or leave comments. www.axiomaticeconomics.com Volume Two, blue and black belt, is not written yet and may not be completed in 2020. Until it is written, I recommend Altshiller-Court’s College Geometry or Johnson’s Advanced Euclidean Geometry for students who have completed my Volume One. The latter is more advanced. Prerequisite for Volume One is Algebra I. No knowledge of non-linear functions, systems of linear equations or set theory is expected. But the chains of reasoning are longer and more difficult than some Algebra I students are ready for. I recommend Hall and Stevens’ A School Geometry for students who need an introduction to geometric proofs before they tackle Geometry–Do. Table of Contents The way that you wander, is the way that you chose, The day that you tarry, is the day that you lose, Sunshine or thunder, a man will always wonder, Where the fair wind blows, where the fair wind blows. If you are coming to this table of contents now for the first time, it may seem as though a black belt in Geometry–Do is a mountain too high to climb. But I tell you, the next three years are going to pass anyway, so why not go for the gold? It is an accomplishment you can boast about for the rest of your life. What else can you do as a teenager that you will be proud of as an old man? Postulates 1 The geometric postulates and the axioms of abstract algebra are explained. Notation 14 White Belt Instruction: Foundations 15 SAS, Isosceles Triangle Theorem and SSS are proven, in that order, to lay the foundation for geometry. Basic constructions (bisecting angles and segments, raising and dropping perpendiculars, replicating angles, etc.) are described. Construction workers are taught how to square foundations and walls working only inside the figure, and how to build gantries, wide gates, bridges, etc. Yellow Belt Instruction: Congruence 37 All the remaining congruence theorems are proven, and every theorem that can be proven without use of the parallel postulate. This organization is for students preparing to take a class in non-Euclidean geometry. The proofs are rigorous; we do not take similarity as an axiom, say “just set dilation to unity” and call it a proof. Architects learn of Gothic arches; they take this on faith while scholars come back to arches after completing orange belt, when they are proven to exist. Orange Belt Instruction: Parallelograms 85 Initially, the parallel postulate is needed because we speak of the intersection of lines and we can only be sure they do in Euclidean geometry. Midway through, it will be explicitly cited for the transversal theorem. There is much problem solving, including minimizing the sum of non-collinear segments and drawing a line through circles so the chords it cuts off meet given conditions. Soldiers learn how to position three guns to triangulate fire on aircraft and to enfilade roads. We conclude with a 20-question multiple-choice green-belt entrance exam. Only 10% survive, comparable to the simsa bout to get a green belt in Tang–Soo–Do. Geometry without Multiplication Victor Aguilar Green Belt Instruction: Triangle Construction 167 This begins second-year geometry, which is an honors-level class. The average students dropped out or failed orange belt and the construction workers quit after yellow belt, but it is hoped that aspiring military officers remain, for there is much to learn about machine gun emplacement. Mariners learn to navigate using lighthouses and structural engineers learn about skew bridges, but most of the applications are for infantry officers, which many boys find interesting even if they do not plan to enlist. For aspiring mathematicians, there is much use of loci and we work through many of the triangle construction problems that are a mainstay of Russian geometry but are rarely considered in the West. Military officers will be pleased that they are learning to fight the Russian way, which is more scientific and makes more use of automatic cannons than NATO does. The positioning of Shilkas is not random; they put a lot of thought into triangulating fire, enfilading roads, and covering possible sniper positions. Red Belt Instruction: Triangles, Advanced 215 By advanced I mean theorems difficult enough that they went unsolved for decades and are now named after famous mathematicians. Volume Two will assume an audience of Olympians; here, while we are still being helpful to engineers and military officers, we are transitioning into helping students compete in the International Mathematical Olympiad. We consider the work of Miquel, Wallace, Torricelli, Napoleon, Fagnano, Euler, et. al. Torricelli’s problem of minimizing the sum of the distances to vertices is solved; and in reverse, to recover the triangle. The Euler segment theorem is proven and the Euler (nine- point) circle is discussed. Homothecy is introduced. Air Force officers learn how to bomb the defense that Army officers designed as orange belts. There are results of interest to military officers that go beyond just laying ambushes. Index of Postulates, Theorems and Constructions 299 Principal Results of Geometry–Do in Alphabetical Order 347 Index of Names 351 References 359 Glossary 363 Glosario Español 373 Das englisch-deutsche Glossar 385 Англо-русский глоссарий 399 Euclid’s Postulates Plus One More Segment Two points fully define the segment between them. Line By extending it, a segment fully defines a line. Triangle Three noncollinear points fully define a triangle. Circle The center and the radius fully define a circle. Right Angle All right angles are equal; equivalently, all straight angles are equal. Parallel A line and a point not on it fully define the parallel through that point. Segments are denoted with a bar, 퐸퐹; rays with an arrow, 퐸퐹⃗⃗⃗⃗⃗ ; lines with a double arrow, 퐸퐹⃡⃗⃗⃗⃗ ; and angles as ∠퐸퐹퐺. The postulates are in terms of fully defined, which means that a figure with the given characteristics is unique, if it exists. Under defined means figures with the given characteristics are legion; more information is needed. John Playfair stated the parallel postulate roughly as I and David Hilbert do, which can be proven to be equivalent to Euclid’s Fifth Postulate. If a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. While Hilbert and I both found Euclid’s postulate to be convoluted and chose Playfair’s version, and we both reject real numbers as unsupported by our postulates, we otherwise took separate paths. Geometry–Do is like Hilbert’s geometry, but it is unique and has its own postulates. Euclid also had five “common notions,” which vaguely describe what modern mathematicians call equivalence relations, total orderings, and additive groups. Equivalence Relations and Total Orderings A relation is an operator, 푅, that returns either a “true” or a “false” when applied to an ordered pair of elements from a given set. For instance, if the set is integers and the relation is equality, then 5 = 5 is true, but 5 = 4 is false. Relations must be applied to objects from the same set. For instance, 퐸퐹 = ∠퐺 is neither true nor false; it is incoherent. There are four ways that relations may be characterized. It is never true that a relation has all four, but some have three. Reflexive 푎 푅 푎 Symmetric 푎 푅 푏 implies 푏 푅 푎 Anti-Symmetric 푎 푅 푏 and 푏 푅 푎 implies 푎 = 푏 Transitive 푎 푅 푏 and 푏 푅 푐 implies 푎 푅 푐 Geometry without Multiplication Victor Aguilar A relation that is reflexive, symmetric, and transitive is called an equivalence relation.
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