Numberphile's "Epic Circles" easy bit

So, I was watching a little bit of Numberphile on YouTube, like you do before you get up on a lazy Sunday.

I was watching the "Epic Circles" video which talks about finding the diameter (or radius) of successive circles in a "Pappus Chain". I've played around in this space, playing with circle inversions, flashing back to a particularly bad math professor for my complex analysis class, but it's all still pretty cool stuff.

22 minutes and 20 seconds into the video, Simon Pampena has something like this that he's working with:

And he makes the claim that the smaller circle has a quarter the radius of the larger circle, so R/4 = n in my diagram. I've clearly redrawn the diagram, and am using R to mean something different than he is, and I've introduced n, which shouldn't be too scary. In any case, what we've got is a square of side R, with a quarter circle of radius R with its center at one corner, and a half circle of radius n tangent to the quarter circle, such that the center of the half circle is on the square and the top of the half circle hits that upper right hand corner. The diagram isn't perfect, but it's close.

In the video, Simon just zips past the claim that R/4 = n and it threw me for a bit - where did that come from? So, I took out some graph paper and doodled a bit. Let's first rename things again, making this into a unit square:

In this picture, I've drawn a line connecting the centers of the two circles. You'll see it goes through the point of tangency. (That's obvious, right?)

So, the radius of the large circle is now 1, and I've renamed the radius of the small circle x, just to make things prettier later on.

One thing I thought about halfway through working it out as I describe below is that maybe there was some well-known relationship going on here, and I did notice this:

That is, if we scale up the square to side length 4, we're asserting that the little circle is now a unit circle, and that makes the red diagonal the hypotenuse of a 3:4:5 right triangle. Ok, that's verifying the claim, but that's not really what I wanted - I wanted to derive x from the previous diagram. And now it occrus to me that x in this diagram is scaled up from x in the previous diagram. Don't get confused, this was a tangent. Sigh, math pun.

Ok, so going back to the second diagram, we have that same right triangle which has the red line as a hypotenuse. The length of the hypotenuse is:

hypotenuse = 1 + x

because it's the sum of the two radii of the circles, 1 and x. The height of the triangle is

height = x - 1

because the height of the square is the radius of the big circle, 1, less the radius of the small circle, x. And the width of the triangle, let's not always see the same hands, is

width = 1

because we said so.

And here's yet another picture of that:

We're nearly done now. All we have to do is hit that triangle with our Pythagoras hammer and we get:

hypotenuse ^ 2 = width ^ 2 + height ^ 2

(1+x) ^ 2 = 1 ^ 2 + (1 - x) ^ 2

1 + 2x + x^2 = 1 + 1 - 2x + x^2

And then we cancel and rearrange. I have to admit that it took me a few tries (I claim it's because I still had that lazy Sunday morning brain on), but if you're careful, you cancel out the x^2 terms, pull the -2x over to the left, pull a one from left to right, and get:

4x = 1

and so

x = 1/4

Which is what we believed all along; the small circle is 1/4 the radius of the big circle, and math works, and I can go back to bed.