Intersecting semi-disks and the synergy of three quadratic forms

In this paper, we study the Diophantine equation x² = n² + mn + np + 2mp with m, n, p, and x being natural numbers. This equation arises from a geometry problem and it leads to representations of primes by each of the three quadratic forms: a² + b², a² + 2b², and 2a² − b². We show that there are infinitely many solutions and conjecture that there are always solutions if x ≥ 5 and x ≠ 7; and, we find a parametrization of the solutions in terms of four integer variables.