\(S_{\Delta AB'C'}=\frac{1}{4}S_{\Delta ABC}\) ; \(d\left(S';\left(ABC\right)\right)=2.d\left(S;\left(ABC\right)\right)\)
\(\Rightarrow V_{S.AB'C'}=\frac{1}{3}d\left(S';\left(ABC\right)\right).S_{\Delta AB'C'}=\frac{1}{3}.2d\left(S;\left(ABC\right)\right).\frac{1}{4}S_{\Delta ABC}\)
\(=\frac{1}{2}\left(\frac{1}{3}d\left(S;\left(ABC\right)\right).S_{\Delta ABC}\right)=\frac{1}{2}V\)