Geometry Beyond Algebra. The Theorem of Overlapped Polynomials (TOP) and its Application to the Sawa Masayoshi\'s Sangaku Problem. The Adventure of Solving a Mathematical Challenge Stated in 1821

TL;DR

This paper solves the 1821 Sawa Masayoshi's problem explicitly for the symmetric case and develops the Theory of Overlapped Polynomials (TOP), demonstrating its potential for broader mathematical applications.

Contribution

It provides the first explicit algebraic solution for the symmetric case and introduces TOP, a new theory with potential for solving complex geometric problems.

Findings

Explicit algebraic solution for symmetric case

Proof of existence and uniqueness for asymmetric case

Development of the Theory of Overlapped Polynomials (TOP)

Abstract

This work presents for the first time a solution to the 1821 unsolved Sawa Masayoshi's problem, giving an explicit and algebraically exact solution for the symmetric case (particular case b = c, i.e., ABC \equiv right-angled isosceles triangle), see (1.60) and (1.61). Despite the isosceles triangle restriction is not necessary, in view of the complexity of the explicit algebraic solution for the symmetric case, one can guessing the impossibility of achieving an explicit relationship for the asymmetric case (the more general case: ABC \equiv right-angled scalene triangle). For this case is given a proof of existence and uniqueness of solution and a proof of the impossibility of getting such a relationship, even implicitly, if the sextic equation (2.54) it isn't solvable. Nevertheless, in (2.56) - (2.58) it is shown the way to solve the asymmetric case under the condition that (2.54) be solvable. Furthermore, it is proved that with a slight modification in the final set of variables (F), it is still possible to establish a relation between them, see (2.59) and (2.61), which provides a bridge that connects the primitive relationship by means of numerical methods, for every given right-angled triangle ABC. And as the attempt to solve Fermat's conjecture (or Fermat's last theorem), culminated more than three centuries later by Andrew Wiles, led to the development of powerful theories of more general scope, the attempt to solve the Masayoshi's problem has led to the development of the Theory of Overlapping Polynomials (TOP), whose application to this problem reveals a great potential that might be extrapolated to other frameworks.