@AkivaWeinberger That reminds me of this XKCD, where they mention that exact thing used as a prank.

I have seen gears that I'm pretty certain can't be written as $r = f(\theta)$, as the teeth are a little bulbous.

@AndrejBauer Because mathematicians have to communicate with one another, and pure, intuitive ideas rarely transfer from one mind to another unchanged. We need a protection against such mutation, and formalism and rigidity is the way the mathematical community has chosen to go. At least that's my two cents.

I would rather say that war moves engineering. Without having any sources to go on other than my gut, I would think it reasonable that if you want a military edge in a war, you don't often go do science and discover new things. You take the things science says ought to be possible, and find ways to actually make it happen in the real world (the Manhattan project comes to mind). Or you take something that exists but is too expensive (or unreliable), and you find ways to make it much cheaper (or sturdier) so that you can supply your army (for instance, air planes during first world war)

@user1998586 I can understand that. I haven't really been able to let this problem go myself. Can't wait until I have some time for myself to sit down and try to prove it, both the two posted, confirmed results and the conjecture we have made here. Alas, that won't be until tomorrow.

@user1998586 As I point out in my comment right above yours, I believe it does, only you have to swap the role of the two pairs of tangent lines. Using the old roles clearly doesn't work, but a quick drawing I made by hand made it seem plausible that it works the other way.

If you draw a circle that is tangent to both of the fixed circles, but circumscribes only one of them, then look at the points where it intersects the two green tangents from the original post (dashed in this picture), would that make two chords that are parallel to the black / solid tangent lines? (I.e. swap the roles of the two pairs of tangent lines.)

"In mathematical terms, you are looking for diagonals in a hypercube…" Or of any of its facets.

@Qfwfq Note that the kernel could be trivial even though the presheaf is not a sheaf; sheafification kills sections which are locally zero, and adds sections which may be constructed locally, and the latter does not contribute to the kernel.