Generalization to higher dimensions of a statement about plane triangles

Question

Let $\Delta=\Delta ABC$ be a plane triangle with area $F_\Delta$ and let $P$ be a point in $\Delta$. Draw lines through $P$ parallel to the sides of $\Delta$; then $\Delta$ is decomposed into three parallelograms and three triangles $\Delta_1$, $\Delta_2$ and $\Delta_3$ (all similar to $\Delta$). For the respective areas $F_1,F_2,F_3$ of these triangles, the formula $$\sqrt{F_\Delta}=\sqrt{F_1}+\sqrt{F_2}+\sqrt{F_3}$$ holds. Is there a generalisation of this to higher dimensions, and what is its precise formulation?