  This article reports the occurrence of binary quadratic forms in primitive
Pythagorean triangles and their geometric interpretation. In addition to the
well-known fact that the hypotenuse, z, of a right triangle, with sides of
integral (relatively prime) length, can be expressed as the sum of two squares,
z=a^2+b^2, where a and b are positive integers of opposite parity such that
a>b>0 and gcd(a,b)=1, it is shown that the sum of the two sides, x and y, can
also be expressed as a binary quadratic form, x+y=(a+b)^2-2b^2. Similarly, when
the radius of the inscribed circle is taken into account, r=b(a-b), a third
binary quadratic form is found, namely (x+y)-4r=z-2r=(a-b)^2+2b^2. The three
quadratic representations accommodate positive integers whose factorizations
can only include primes p represented by the same type of binary quadratic
forms, i.e. p=1,5(mod8), p=1,7(mod8), and p=1,3(mod8), respectively. For all
three types of binary quadratic forms, when the positive integers represented
are prime, such representations are unique. This implies that all odd primes
can be geometrically incorporated into primitive Pythagorean triangles.
