SOLVING SOME GEOMETRIC PROBLEMS MORE EFFICIENTLY USING GENERALIZED PTOLEMY’S THEOREM
DOI:
https://doi.org/10.18623/rvd.v22.n4.3701Palavras-chave:
Ptolemy’s Theorem. Generalized, Ptolemy’s Theorem, Problem SolvingResumo
If four points in this order, lie on a circle , there is an interesting and useful result, called Ptolemy’s Theorem, which links the lengths of the legs of the quadrilateral with the lengths of its diagonals, by the equality This result may be generalized if the points are replaced by circles that touch the circle The result is known as Generalized Ptolemy’s Theorem (GPT), also known as Casey’s Theorem. This relatively less-known result may be used to prove several geometric results. In this paper, we apply GPT to partially known special problems, demonstrating the efficiency of this proving tool. We also compare it with other ways of proving the results, omitting the GPT. Finally, we also expect that one can pose similar geometric problems, where GPT can efficiently be used as a solving tool.
Referências
Gueron, Sh. (2002). Two Applications of Generalized Ptolemy Theorem. The American Mathematical Monthly, vol. (109), No. 4, 362-370.
Johnson, R. A. (1960). Tangent Circles. In J. W. Young (Ed.), Advanced Euclidean Geometry (pp. 121-126), Dover Publications, Inc., New York.
Weisstein, E. W. (1998). Casey's Theorem. From MathWorld-A Wolfram Resource. https://mathworld.wolfram.com/CaseysTheorem.html
Larson, C. L. (1983). Heuristics, Geometry. In P. R. Halmos (Ed.), Problem-Solving Through Problems (pp. 290-291), Springer-Verlag, New York, Inc.
Shirali, Sh. (1995). On The Generalized Ptolemy Theorem. From https://paperzz.com/doc/7217117/on-the-generalized-ptolemy-theorem.
Gonzalez, L. (2011). Casey’s Theorem and its Applications. From https://geometry.ru/articles/Luis_Casey.pdf.
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