Geometry: given a fixed area and vertex angle, prove that an isosceles triangle would minimize the length of the base

Question

I was working on this problem and could intuitively see how this is true. I tried to solve it on my own, but the only "proof" I could come up with was that because the non-base sides increase at different rates in order to keep the area the same, the base length must increase in order to close the triangle. Is there a more formal proof of this problem?