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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