# Solid geometry /Volumes

I Need Some Help About The Geometry Problem, Question : " Lets suppose that Lines l and l' and l'' have cut each other in point A; The B And B' Are two random points From Line l , C and C' Are random points in line l' and D and D' Are Random Points From Line l " How to Proof This equation : $$\frac{V_{ABCD}}{V_{AB'C'D'}} = \frac{\vert AB \vert \times \vert AC \vert \times \vert AD \vert}{\vert AB' \vert \times \vert AC' \vert \times \vert AD' \vert}$$

And We suppose that we can't use the trig laws.

Draw the height $AH$ and $A_1H_1$. $$AH||A_1H_1 \Rightarrow \triangle ASH \sim \triangle A_1SH_1 \Rightarrow$$ $$\Rightarrow \frac{A_1H_1}{AH}=\frac{SA}{SA_1}$$ and $$\angle C_1SB_1 = \angle CSB = \alpha.$$ $$\frac{V_{SA_1B_1C_1}}{V_{SABC}}=\frac{A_1H_1\cdot S_{SC_1B_1}}{AH\cdot S_{SCB}}=\frac{SA_1}{SA}\cdot\frac{\frac12 SB_1\cdot SC_1 \sin \alpha}{\frac12 SB\cdot SC \sin \alpha}=$$ $$=\frac{SA_1\cdot SB_1 \cdot SC_1}{SA\cdot SB \cdot SC}$$