Left as Exercise

Last post ended up way longer than expected with the backstory, so posting a small joke to satisfy potentially tired readers (and a lesson I learnt from the situation).

I was studying some 3D geometry by following a Chinese math class yesterday, and there was a problem used to introduce a concept.  Basically, there was this situation, where a line $\ell$ that is part of plane $\mathcal Q$.  $\ell$ is parallel to another plane $\mathcal P$, and $\mathcal P \perp \mathcal Q$.  Let line $k=\mathcal P\cap \mathcal Q$, and trivially, $\ell\parallel k$...

(The proof is that if $\ell \not\parallel k$, then they'd intersect, and that intersection would be on $k$ and thus also on $\mathcal Q$, which violates $\ell\parallel\mathcal Q$.)

There were more entities and constraints involved in the problem, but I just wrote "trivially, $\ell\parallel k$" as my first line in my notes.  And while it was likely citable as trivial, I wasted 40 minutes proving this "trivial" thing today.  Insert the classic "left as exercise" meme here, except it was used against myself.

Now, you may see me as a stupid guy who had no idea what he was doing, as this is a really fundamental property.  The truth is, I spent most of the 40 minutes reasoning with the other conditions in this problem, and didn't even recognize that the conditions I've mentioned above are present in the problem.  Once you know what you need to do, it's easy to see whatever the heck is going on and produce a good argument for this "trivial" statement.  But you need to know where it comes from.

But if I were familiar with this fact in the first place, I'd know to look for those conditions directly.  I guess this is why we need to memorize common theorems and facts for not just math, but every subject, despite how much I dread (yet am surprisingly good at, according to some people; I don't know personall) memorization.  But also, I'm probably going to watch out for using the word "obviously" or "trivially" in the future.