This semester I am taking an introductory course on projective geometry. Even though I believe I understand the different theoretical concepts correctly (supplementary varieties, Grassmann Formula, etc.) more often than not I find myself completely baffled when trying to attempt different problems.
For instance, one I was working on recently was:
"In a projective space of dimension four, $\mathbb{P}_4$, let $\pi_1, \pi_2$ be two planes that intersect at point, $\pi_1 \cap \pi_2 = \{p\}$, and let r be a line that passes through $p$ and that isn't contained in $\pi_1$ nor in $\pi_2$. Prove that there exists a unique plane $\pi$ that contains $r$ and s.t. the intersections $\pi \cap \pi_1$, $\pi \cap \pi_1$ are lines."
My problem is that when faced with problems like this I don't know how to start. Since we are working in a dimension higher than 3, I don't have any kind of geometrical intuition (for instance, if we were in dimension 3 I believe I could think of $\mathbb{P}_3$ simply as an affine space of dimension 3, completed with points at infinity). I have seen the solution to the problem, in which the main idea is to think about a supplementary variety to $p$, but I don't know how someone would come to think of that. So, my question is, how should one tackle this kinds of problems?
Also, any recommendations for a projective geometry book in which explicit, step-by-step solutions to problems are given would be very much welcome.