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This article reports the occurrence of binary quadratic forms in primitivePythagorean triangles and their geometric interpretation. In addition to thewell-known fact that the hypotenuse, z, of a right triangle, with sides ofintegral (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 thata>b>0 and gcd(a,b)=1, it is shown that the sum of the two sides, x and y, canalso be expressed as a binary quadratic form, x+y=(a+b)^2-2b^2. Similarly, whenthe radius of the inscribed circle is taken into account, r=b(a-b), a thirdbinary quadratic form is found, namely (x+y)-4r=z-2r=(a-b)^2+2b^2. The threequadratic representations accommodate positive integers whose factorizationscan only include primes p represented by the same type of binary quadraticforms, i.e. p=1,5(mod8), p=1,7(mod8), and p=1,3(mod8), respectively. For allthree types of binary quadratic forms, when the positive integers representedare prime, such representations are unique. This implies that all odd primescan be geometrically incorporated into primitive Pythagorean triangles.

The Triangular Theorem of the Primes: Binary Quadratic Forms and Primitive Pythagorean Triples