THEOREM OF PYTHAGORAS. FOUR NEW PROOFS

Abstract. Four new proofs of Pythagoras' theorem are given, the first two of which are derived from the similarity of triangles, and the last two from the similarity of triangles and the calculation of the areas of triangles. Unlike the well-known proofs of the Pythagorean Theorem, the last two proofs were reduced to the case where the product of two algebraic terms is zero. Equating the first term to zero reduced the proof to the general case of the Pythagorean Theorem, and the second term to the special case, which is easy to prove. The Pythagorean Theorem is a good example for the mathematical education of pupils and students, as the number of proofs is not limited. Proofs of this theorem in different ways are very good algebro-geometric exercises. Schools and universities can organise competitions for the greatest number of proofs of the Pythagorean Theorem. These competitions could lead to new proofs of the theorem. All this could develop into a kind of movement called "Pythagoreana", which would be very useful in raising the prestige of mathematical education among young people. This article is a translation of a Russian-language article published under the title “The Pythagorean Theorem. For new evidence” in Scientific Bulletin of Belgorod State University. No. 20 (241), Iss.44, 2016, pp. 34 – 41. https://core.ac.uk/download/pdf/151231313.pdf (In Russian).