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A generalized Pythagorean theorem is an equation relating the squares of the volumes of faces of a particular k-simplex in n-dimensional Euclidean space. There are many proofs of this theorem. This...

We prove a necessary and sufficient condition for a given integer to be the hypotenuse of a primitive Pythagorean triangle. This characterization is based on partitions into consecutive parts.

We present a parametrization for the Pythagorean matrices, i.e., integer-valued matrices A, B, and C such that A2+B2=C2 .

To do this, we show that it is possible to introduce simple ^-vectors?those of the form ax a A ak where ax,... , ak are vectors in some Rn?as geometric objects. In calculus or an introductory physics class, one sometimes hears a vector described as an equivalence class of directed line segments. It turns out that one can, in a similar way, define a simple /c-vector as an equivalence class of oriented parallelepipeds. We shall show that this geometric definition leads to the standard properties of exterior algebra. To illustrate the use and convenience of this machinery, we then generalize the triangle law of vector addition and the law of cosines to ft-simplexes and rederive the ft-dimensional versions of the parallelogram law and the Pythagorean theorem.

(2006). The Pythagorean Theorem: What Is It About? The American Mathematical Monthly: Vol. 113, No. 3, pp. 261-265.

Planar triangles with each side having a positive integer length are among the simplest geometrical objects imaginable. What can be said about the angles of such triangles? In particular, are these angles rational or at least algebraic numbers when measured in degrees? This article demonstrates that the answer in general is negative, except for three distinguished families of triangles. One family is well known since antiquity: With the largest angle equal to 90 degrees, it simply is the family of Pythagorean right triangles. Though not nearly as well-known, the other two families also deserve to be part of every geometry teacher's toolkit.

(2016). A Proof of The Pythagorean Theorem After Descartes. The American Mathematical Monthly: Vol. 123, No. 4, pp. 386-386.

(2001). Pythagoras' Theorem for Areas. The American Mathematical Monthly: Vol. 108, No. 6, pp. 549-551.

ALBERT FASSLER obtained his bachelor's degree in Mechanical Engineering at the HTL in Winterthur before he studied Mathematics at the Eidgenossische Technische Hochschule ETH in Zurich (Federal Institute of Technology). After graduation, he worked under Eduard Stiefel at the Seminar ffur Angewandte Mathematik (Seminar for Applied Mathematics) at the ETH, where he got his Ph.D. and became a lecturer. Together with Stiefel, he wrote a book in the field of applied group theory with the title Gruppentheoretische Methoden und ihre Anwendung (Teubner, Stuttgart, 1979). He is now at the HTL Biel/Bienne (an Engineering college).

(2015). A New Proof of the Pythagorean Theorem. The American Mathematical Monthly: Vol. 122, No. 5, pp. 451-451.

We use a 200-year-old theorem on determinants to prove a very general version of the Pythagorean theorem. It relates the square of the n-dimensional volume of an n-parallelotope in ℝm to the sum of...

Suppose that a 2 C b 2 D c 2 , where a, b, and c are relatively prime positive integers, and consider the right triangle T with sides a, b, and c. We prove that both of the acute angles in T can be trisected with a compass and unmarked straightedge if and only if c is a perfect cube. 1. INTRODUCTION. It has been known for about two centuries that some angles cannot be trisected using only a compass and straightedge (assuming the rules es- tablished by the classic Greek geometers). Most sources mention that certain angles such as 90 , 45 , and 9 can be trisected with only these tools and then prove that a 60 angle cannot be trisected in this way. In this note, we determine necessary and sufficient conditions for both of the acute angles in right triangles corresponding to Pythagorean triples to be trisectible with only a compass and straightedge. Since both the trisection of angles problem and the properties of Pythagorean triples have been studied extensively, we leave most of the background details (which are tedious and distract from our primary goal) to the reader. There are a plethora of references for these subjects; the short list provided at the end of this note barely scratches the surface on these topics. A primitive Pythagorean triple.a;b;c/ consists of three relatively prime positive integers such that a 2 C b 2 D c 2 , while a primitive Pythagorean triangle refers to the corresponding right triangle having side lengths of a, b, and c. It is easy to verify that c must be odd and that a and b must have oppo- site parity; we will always assume that the first integer in the triple is even. Note that primitive Pythagorean triangles (and thus their acute angles) can be constructed with a compass and straightedge, since the sides of the triangle have integer lengths. 2. PRELIMINARIES. To prove our main result (Theorem 3), we need two theo- retical results. We encourage the reader to look into the details behind each of these facts. The first result involves the theory of constructible numbers. A number is con- structible if, given a line segment of length x, a line segment of lengthj jx can be constructed with a compass and straightedge. It can be shown that every constructible number is an algebraic number for which the degree of its minimal polynomial is a power of two. See (1) or (6) for a proof of this fact. The geometry text (8) does a nice job presenting an informal discussion of constructible numbers and explaining why a 60 angle cannot be trisected. Theorem 1. Suppose that cos D r, where r is a rational number. The angle can be trisected with a compass and straightedge if and only if the polynomial 4x 3 3x r

Novel proofs of the Pythagorean Theorem are obtained by dissecting the squares on the sides of the abc triangle into a series of infinitely many similar triangles.

1. INTRODUCTION. We often fool our students. We fool them by stating and proving theorems that are in a polished and final form. As a result, students are often unaware of the evolution of ideas, strategies, statements, proofs and even important mistakes that inevitably lead to the beautiful theorems and proofs they encounter. There is much value in understanding early ideas and mistakes. In some cases, mistakes lead to the discovery of powerful and deep new mathematical truths. Thus we believe that it is occasionally appropriate to celebrate mathematical mistakes and we cannot imagine a better time for such a celebration than on April Fools' Day. Here we describe and outline three erroneous proofs from the l9th century. We provide "proofs" of Fermat's Last Theorem, the Four Color Conjecture, and the fact that one of us is Bill Clinton. This paper is based upon a special undergraduate mathematics colloquium the authors gave on April 1, 1996. 2. FERMAT'S LAST THEOREM. The problem that was to become one of the most infamous open problems in mathematics had a most inauspicious beginning. Circa 1637, Pierre de Fermat (Figure la), while studying Bachet's Latin translation of Arithmetica by Diophantus, came upon a discussion of the Pythagorean theorem. This inspired Fermat to write the following, now famous, lines in the margin

Since Pythagoras, philosophy of mathematics tried to account for mathematical existence and the nature of mathematical objects. Numbers, circles, n-dimensional manifolds, all are different from everything else we think about. They're neither physical nor mental. Not mental, because the Pythagorean theorem or any other well-established mathematical fact is independent of what you or I think. Whether we know it and believe it or don't know it and don't believe it, the Pythagorean theorem is still true. Yet it's not physical either! Plato and Aristotle explained that the triangles and circles of the geometer are not physical triangles or circles, but something "ideal." Spiritual, empirical, psychological, formalist, and logicist explanations have been offered. None give a credible account of what we do when we do mathematics. Presently some authors are constructing a humanist answer.

(1938). Pythagorean Sets of Numbers. The American Mathematical Monthly: Vol. 45, No. 5, pp. 298-301.

(1897). New and Old Proofs of the Pythagorean Theorem. The American Mathematical Monthly: Vol. 4, No. 3, pp. 79-81.

where I=2(p-qb), m=2(p-gqa), a= l (1 + \15), b-(1-\/5). The purpose of this article is to find a connection between generalized Fibonacci numbers and Pythagorean number triples. By a Pythagorean (number) triple is meant a set of three mutually prime integers u, v, w for which u2+v2 = w2. The problem to be solved is this: Given such a triple u, v, w, can we find n, p, q such that the integers whose squares appear in (3) below are these u, v, w? The answer is yes. Viewed in this light, Pythagorean triples may be called Fibonacci (number) triples.

A Pythagorean triangle is a right triangle with integer sides. A well-known example is the triangle with sides 3, 4, and 5. Unfortunately, the angles of these triangles are not as "nice" as the sides: whether expressed in degrees or radians, they are always irrational. If a Pythagorean triangle has an angle of A degrees and IA 201 < 1/100, we shall say that the triangle "approximates 20 degrees to within one hundredth." Let B be any given angle in degrees with 0 < B < 90. Let e be any real number such that 0 < e < 1, e < B, and e < 90 B. The question we answer in this note is: what is the smallest Pythagorean triangle which approximates B degrees to within

(1979). The Curious Substitution z = tan θ/2 and the Pythagorean Theorem. The American Mathematical Monthly: Vol. 86, No. 7, pp. 584-585.

(1977). A Converse of Pythagoras' Theorem. The American Mathematical Monthly: Vol. 84, No. 7, pp. 551-553.

(1974). Generalized Pythagorean Theorem. The American Mathematical Monthly: Vol. 81, No. 3, pp. 262-265.