On the Pythagorean Theorem - Issues of Analysis

On the Pythagorean TheoremA. V. Petrov2010 Mathematical Subject Classification 26A21.AbstractThe relation between the lengths of the right-angled trianglesides is introduced in this paper.Key words: right-angled triangle, Pythagoras.1 The Pythagorean TheoremWe need the following lemma.Lemma 1 The square of the side length equals to the area of the squareconstructed on this side.Proof.This assertion is evident.✷Theorem 1 In a right-angled triangle the hypotenuse square equals to thesum of the legs squares.Proof. Summing the area of the squares on the legs we obtain the areaof the square on the hypotenuse. ✷Remark 1 The equalitya 2 + b 2 = c 2in the Pythagorean theorem is the necessary and sufficient condition for atriangle to be right-angled.For the details see [1].1

References[1] Wirsing E. Das asimtotische Verhalten von Summen uber multiplikativeFunktionen. II // Acta Math. Sci. Hung. V. 18. 1967. P. 411–467.[2] Varadarjan V. S. Harmonic analysis on real reductive groups. Berlin;Heidelberg; New York: Springer-Verlag, 1977.Petrozavodsk State University,Petrozavodsk, Lenin Avenue, 33.E-mail: petrov@petrsu.ruAbstract. The Pythagorean theorem is proved in this paper.2