A Combinatorial Proof for Partitions of Pythagorean Triples Into Three Parts

Any Pythagorean triple $\{a,b,c\}$ such that $a^2+b^2=c^2$ satisfies an elegant relation between its partitions into three parts, namely $p(a,3)+p(b,3)=p(c,3)$. While this property follows from elementary analytic methods, we give the first combinatorial proof of this relation.