# Show that these 3 projective lines intersect on the same point

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!

## 1 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 $$\overline{P_3S_3}$$.