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This work presents for the first time a solution to the 1821 unsolved SawaMasayoshi's problem, giving an explicit and algebraically exact solution forthe symmetric case (particular case b = c, i.e., ABC \equiv right-angledisosceles triangle), see (1.60) and (1.61).  Despite the isosceles triangle restriction is not necessary, in view of thecomplexity of the explicit algebraic solution for the symmetric case, one canguessing the impossibility of achieving an explicit relationship for theasymmetric case (the more general case: ABC \equiv right-angled scalenetriangle). For this case is given a proof of existence and uniqueness ofsolution and a proof of the impossibility of getting such a relationship, evenimplicitly, 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 thecondition that (2.54) be solvable. Furthermore, it is proved that with a slightmodification in the final set of variables (F), it is still possible toestablish a relation between them, see (2.59) and (2.61), which provides abridge 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 thedevelopment of powerful theories of more general scope, the attempt to solvethe Masayoshi's problem has led to the development of the Theory of OverlappingPolynomials (TOP), whose application to this problem reveals a great potentialthat might be extrapolated to other frameworks.

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