So we’re off for Thanksgiving today (phew!) and after a few really great days, this two-day week was pretty much a bummer. The one fun mathematical thing I’ve worked on is a problem involving geometry. Ew, I know. But I got really into it, and it got me thinking of about a million other extensions, questions, methods of attack, etc. I was thinking in so many different ways — about symmetry and about limiting and degenerate cases and about angles and such.
(1) You have two circles, radius 3 and radius 5, tangent to each other. You want to draw a third circle tangent to the given two circles. In fact, you realize there are an infinite number of these circles. So the first question is: what is the locus of all points which is comprised of the centers of these infinite circles?
If you want a small hint, go after the jump for a picture. (In case it wasn’t clear, we are taking about circles which are externally tangent to each other.)
(2) Generalize the problem to being given two circles with radius a and radius b (instead of radius 3 and radius 5).
(3) Can you find a third circle tangent to the given two circles such that the centers of the three circles forms a right triangle (if possible).
(4) What if we ask a similar question about spheres? If you are given two spheres of radius a and radius b, what is the locus of all points which is comprised of the centers of spheres tangent to the given two spheres?
So if you are bored over your Thanksgiving holiday, you might want to have some fun with this. I’ve solved the first two. I haven’t had time to think about the third yet, though I know the solution won’t be (too) hard. The fourth one? Eh, I anticipate it to be pretty tough. But having solved the first two will definitely help! [update 5 minutes after posting: Eh, nevermind, I think I know the answer to the fourth one… not really hard!]