Show that these 3 projective lines intersect on the same point

algebraic-curvesprojective-geometry

I'm stuck with the following problem of projective geometry from an assignment:

Let $P_1$, $P_2$, $P_3$ and $Q$ be four points in the projective plane (over an algebraically closed field) such that any 3 of them are not collinear. Define
$Q_1=\overline{P_1Q}\cap\overline{P_2P_3}$, $Q_2=\overline{P_2Q}\cap\overline{P_1P_3}$ and
$Q_3=\overline{P_3Q}\cap\overline{P_1P_2}$. Now, let $R$ be another point not contained on the lines $\overline{P_1P_2}$, $\overline{P_1P_3}$ and $\overline{P_2P_3}$ and define $R_1$, $R_2$ and $R_3$ the same way as above.

Let $\alpha_1$ be the only automorphism of the line $\overline{P_2P_3}$ such that $\alpha_1(Q_1)=Q_1$, $\alpha_1(P_2)=P_3$ and $\alpha_1(P_3)=P_2$. Define $S_1=\alpha_1(R_1)$. Similarly, define $S_2$ and $S_3$. Show that the lines $\overline{P_1S_1}$, $\overline{P_2S_2}$ and $\overline{P_3S_3}$ intersect in the same point.

Since the statement is a bit messy, I tried to make a figure to help understanding it (which is also a bit messy):

I'm trying to use cross-ratio to show that the points $\overline{P_1S_1}\cap\overline{P_2S_2}$ and $\overline{P_2S_2}\cap\overline{P_3S_3}$ are equal: I would express the cross-ratio between one of this points and 3 other (fixed) points of the line $\overline{P_2S_2}$ and, using the invariance properties of the cross-ratio, express it in terms of the other know points mentioned above, to show that the cross-ratio between the other point and the 3 fixed points are the same. But none of my attempts of finding adequate central projections helped me with that, the cross-ratios obtained would always involve "strange points" for which I don't know anything (i.e. points other than those mentioned above).

Could someone please help me with a hint (not a full solution, since it's an assignment)? Thanks!

Best Answer

The problem is a lot simpler than it looks and I was able to figure out the solution thanks to a tip given by the instructor. The tip is:

One can assume, via projective transformation, that $P_1=(1:0:0)$, $P_2=(0:1:0)$, $P_3=(0:0:1)$ and $Q=(1:1:1)$.

With this, one is able to express $S_1$, $S_2$ and $S_3$ simply in terms of $R$ and calculate directly the intersection between the three lines $\overline{P_1S_1}$, $\overline{P_2S_2}$ and $\overline{P_3S_3}$.

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