  From Euclid's fundamental formula for the Pythagorean triples we define the
rational triples relating certain congruent numbers by an identity and explore
their relationships. We introduce two geometric methods relating the congruent
number problem to pairs of conic sections. We show a relationship between the
Cassini ovals and the congruent number problem. By the tangent method we define
a set of rational triangles from an initial solution for a congruent number. We
define the prime footprint equations for right triangles for certain congruent
numbers. By the unseen recurrence we define infinite trees of rational
triangles and congruent numbers. Congruent number families are defined related
to the Fibonacci/Lucas numbers and the Chebyshev polynomials. We show that the
semiperimeter of the Brahmagupta triangles are congruent numbers in function of
the Chebyshev polynomials of the first kind. We present a naive recursive
algorithm to solve the problem posed by Fermat for triangles with $a+b=\square$
and $c=\square$
