PROOF OF THE GENERALIZED PYTHAGOREAN THEOREM IN THE TRI-RECTANGULAR TETRAHEDRON VIA GRAM MATRIX
Keywords:
Generalization of the Pythagorean Theorem, Gram Matrix, Dot ProdutctAbstract
This work presents another way to generalize the Pythagorean theorem in three-dimensional space, using a tri-rectangular trihedron that forms a tetrahedron. The goal of this research is to demonstrate that the square of the area of the opposite face is equal to the sum of the squares of the areas of the other faces in a trihedron. For this purpose, concepts of dot product, Lagrange Identity, and Gram matrix were employed. The results showed that in its various applications, it leads to a series of important results and conclusions in Mathematics, Engineering, Science, and Geometry. Thus, it is concluded that the generalization of the Pythagorean theorem by a tri-rectangular trihedron using the Gram matrix and this Important Lagrange Identity is an important contribution, as it extends its relevance and importance in mathematics and other areas of knowledge.
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Copyright (c) 2025 Ivanildo Silva Abreu, Jean Robert Pereira Rodrigues, Adson Pinto Nunes, Paulo Victor Brito Araújo, Alan Jefferson Lima Aragão, Isaias dos Santos Penha, Cristovam Filho Dervalmar Rodrigues Teixeira, José Ribamar Ferreira Silva, Marlos Luís Rocha Martins, Lourival Matos de Sousa Filho
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
