VI.
[tex]\it 2AB=3BC=4AC \Rightarrow \begin{cases} \it 2AB=4AC|_{:2} \Rightarrow AB=2AC \\ \\ \it 3BC=4AC \Rightarrow BC=\dfrac{4}{3}AC \end{cases}\\ \\ \\ \mathcal{P} =AB+AC+BC=52 \Rightarrow 2AC+AC+\dfrac{4}{3}AC=52|_{\cdot3} \Rightarrow \\ \\ \\ \Rightarrow 6AC+3AC+4AC=3\cdot52 \Rightarrow 13AC=3\cdot52|_{:13} \Rightarrow AC=3\cdot4=12\ cm[/tex]
[tex]\it AB=2AC=2\cdot12=24\ cm\\ \\ BC=\dfrac{4}{\not3}\cdot\ \not12\ ^4=4\cdot4=16\ cm[/tex]