0
$\begingroup$

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!

$\endgroup$
0
$\begingroup$

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 $\overline{P_3S_3}$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.