$a+b\mid c^2 \implies a+b\mid (a+b+c)^2$, combining all the relationships of this kind we get $$(a+b)(a+c)(b+c)\mid (a+b+c)^2,$$implying $$(a+b)(a+c)(b+c)\leq (a+b+c)^2.$$On the other hand, the inequalities derived directly from conditions in b implies $a, b, c \geq 2$ and hence we have $$b^2c+c^2b \geq 2(b^2+c^2) >\frac{b^2}{2} + \frac{c^2}{2} + 2bc.$$Summing up such inequalities we have $$(a+b)(b+c)(c+a)>(a+b+c)^2.$$The contradiction implies the desired result.