Maybe an impossible shape?

Geometry Level pending

There is a cone, a hemisphere and a cylinder standing on an equal base. Given that they have same heights and the heights are equal to the radius, their volumes would have a definite ratio.

If the ratio is $$a : b : c$$, where $$a,b,c$$ are positive integers such that $$a+b+c$$ is minimized, find $$ab + bc - ac$$.

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