Skew Quads

We can rewrite our expression as $\frac{AP}{SA} \cdot \frac{BQ}{PB} \cdot \frac{CR}{QC} \cdot \frac{DS}{RD}$ .

Then, we use the law of sines in triangles $APS$ , $BQP$ , $CRQ$ , $DSR$ , that says that, particularly in triangle $APS$ , $\frac{AP}{\sin \angle ASP} = \frac{SA}{\sin \angle SPA}$ We then use that to write $\frac{AP}{SA} = \frac{\sin \angle ASP}{\sin \angle SPA}$ .

Using this in all the triangles mentioned before, our expression is now $\frac{\sin \angle ASP}{\sin \angle SPA} \cdot \frac{\sin \angle BPQ}{\sin \angle PQB} \cdot \frac{\sin \angle CQR}{\sin \angle QRC} \cdot \frac{\sin \angle DRS}{\sin \angle RSD}$

Now, we rearrange once again to obtain $\frac{\sin \angle ASP}{\sin \angle RSD} \cdot \frac{\sin \angle BPQ}{\sin \angle SPA} \cdot \frac{\sin \angle CQR}{\sin \angle PQB} \cdot \frac{\sin \angle DRS}{\sin \angle QRC}$ Since $P$ , $Q$ , $R$ and $S$ are coplanar, the angles mentioned are the angles formed by each side of the quadrilateral $ABCD$ and the plane $PQRS$ . This means that, as a particular example, $\angle ASP = \angle RSD$ (this is because the angle formed by a line $L$ not in a plane $\pi$ and the plane is the same as the one between $L$ and every line in $\pi$ ; so the angle formed by $AS$ and $SP$ is congruent to the one between $AS$ and $RS$ ). This means that $\frac{\sin \angle ASP}{\sin \angle RSD} = 1$ , and if used with every pair of angles, it makes $\frac{\sin \angle ASP}{\sin \angle RSD} \cdot \frac{\sin \angle BPQ}{\sin \angle SPA} \cdot \frac{\sin \angle CQR}{\sin \angle PQB} \cdot \frac{\sin \angle DRS}{\sin \angle QRC} = 1$ Thus, $\boxed{\frac{AP}{PB} \cdot \frac{BQ}{QC} \cdot \frac{CR}{RD} \cdot \frac{DS}{SA} = 1}$

:P Just draw perpendiculars from each point of the skewed quadrilateral to the plane, then the ratios in question are transformed into ratios of heights, which, will multiply one by succesive cancellation.

The easiest solution for the placement of P, Q, R, and S is the midpoint of the 4 sides. Thus each fraction equals 1, producing a final answer of $\boxed{1}$ .

For the points to be coplanar, the line between one pair would have to be parallel to the line between the other pair. Say we decide to put P and Q together as a pair. There would have to be a horizontal line connecting P and Q, meaning that PB=BQ. Therefore, AP=QC. This means that when we multiply the 2 fractions at the beginning of the equation, both sides cancel out to get 1. Same logic applies for the other 2 fractions/points, and then you get 1*1=1. So 1 is the answer!