Don't trip

Let $$S$$ be the set of all distinct Pythagorean triples $$(a,b,c)$$, where $$a,b,c$$ are positive integers with $$a \lt b \lt c$$, such that $$ab = 6(a + b + c).$$

Find the sum of all distinct values for $$c$$ of the elements of $$S.$$

×