  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.
