I'm reading through a history of math book and it says to
$(i)$ Interpret the figure as an illustration of Pappus’s theorem
$(ii)$ Write down a statement of the theorem corresponding to the figure, the conclusion of which is that $P_1Q_3$ and $P_2Q_2$ are parallel.
$(iii)$ Deduce the required equation from two other equations that express parallelism in the figure.
$(iv)$ Prove the theorem in the case where the two lines $P_1P_2$ and $Q_1Q_2$ do not meet at $O$, that is, when they too are parallel.
$(i)$ My book says that the usual statement of Pappus’s theorem is that
the intersections of opposite sides of the hexagon are in a straight line.
so I'm not sure how I would interpret the figure as an illustration of Pappus's theorem.
$(ii)$ I think the theorem statement would be that $P_1Q_3$ and $P_2Q_2$ are parallel if and only if $$\frac{OP_1}{OP_2}=\frac{OQ_3}{OQ_2}$$
$(iii)$ In an attempt to prove this, I have assumed parallelity of $P_1Q_1$ and $P_3Q_2$ implying $\frac{OP_1}{OP_3}=\frac{OQ_1}{OQ_2}$
I have also assumed parallelity of $P_2Q_1$ and $P_3Q_3$ implying $\frac{OP_3}{OP_2}=\frac{OQ_3}{OQ_1}$
From here I went with a reverse proof:
$$\begin{align*} \frac{OQ_3}{OQ_2} &=\frac{\frac{OP_3}{OP_2}OQ_1}{\frac{OP_3}{OP_1}OQ_1}\\\\ &=\frac{OP_1}{OP_2}\\\\ &\Rightarrow \frac{OP_1}{OP_2}=\frac{OQ_3}{OQ_2} \end{align*}$$
$(iv)$ I'm not quite sure what it is asking or how to approach this one.
I'm looking for insight into $(i)$ and $(iv)$ and whether or not I did $(ii)$ and $(iii)$ correctly.
(i) My book says that the usual statement of Pappus’s theorem is that the intersections of opposite sides of the hexagon are in a straight line. so I’m not sure how I would interpret the figure as an illustration of Pappus's Theorem.
If the two original lines are parallel then O is sitting out there on the line at infinity as well, so all four points (O and the three intersections of the opposite sides of the hexagon) are colinear. Looks like this: