Now let's draw three circles, generic circles that are separated like this in space.
Given a pair of circles,
for example, large and middle,
you can draw a common tangent to those two circles
like so and like so.
To be precise we are drawing two outer tangents because you can
cross these like this, a tangent and tangent so they are inner tangents that cross,
but let's talk about the outer tangents.
And those two outer tangents, these circles are usually of different sizes,
will meet at the vertex or tip which I'll mark in black dot.
And because there are three pairs: large and middle,
middle and small,
and small and large;
there are three pairs of tangents.
And the theorem is that if you take the tip of the first pair, tip of the second pair
of tangents and tip of the third pair of tangents;
they are all aligned along a straight line.
What are the chance that three arbitrary points on a plane
are aligned along a straight line?
Zero.
But in this case, they do align very, very beautifully along a straight line.
This theorem goes back, as far as I know,
to a French mathematician-engineer called Monge, one of the founders of École Polytechnique in Paris.
And he is very famous for descriptive geometry and "remblais" and "déblais",
so how to move a pile of stuff from here to there,
minimising energy and cost and that kind of research.
But he also has a few contributions in elementary geometry like this. And this is,
um, provable by different methods, for example Menelaus and so forth
will do it in a classical method; but there's a very, very beautiful approach,
which again has to do with lifting the whole picture to three dimensions.
Instead of those three circles, let's imagine that these are actually
pictures of three spheres as seen from above.
Please imagine that those three spheres are sitting on the same table.
In other words, their centres are at different heights.
Large one will be lifted a lot
and the small one will be lifted only a small amount, and middle one a little middle-ish.
But they're all sitting on the same table.
Given, say, a large ball and a middle ball,
you can draw a common tangent cone.
So a cone like an ice cream cone, so something that-
a surface like this which sort of expands,
and which sort of shrinks, shrinks, shrinks to a tip.
And if you look at that picture from above,
you see what we saw before, that is, two circles and pair of common tangent lines
and a tip.
So you can interpret that picture as actually
a three-dimensional picture, but seen in two-dimensional appearance, okay.
And because there are three balls;
large and middle will have a cone and tip,
middle and small will have a cone and tip,
and small and large will have a cone and tip;
you have three tips of those three cones.
On the other hand,
please imagine again those three spheres sitting on a common floor,
and now, if you please, take a transparent ceiling from above,
and just tilt it, tilt it and just lower it
so that it touches those three spheres at the same time from above. You have to tilt it, yeah?
It can't be horizontal, you have to tilt it, and it touches those three spheres at the same time.
So you have now a ceiling
which touches those three spheres from above,
and the floor which touches those three spheres from below.
Each of those cones that we talked about,
for example this cone, between large and middle sphere,
well, it's touching this ceiling and the floor at the same time as this cone.
So the tip of this cone is
inside the ceiling and inside the floor at the same time,
okay. And the same goes for the other two tips.
So all those tips of the cones
are in the top plane that is the ceiling, and the bottom plane that is floor, at the same time.
But you know, those two planes in space
intersect along a straight line.
So we have just shown that those tips of cones are along a straight line, and what is that straight line?
It's the intersection of the ceiling that we put from above and the floor on which we
put the balls. And if you look at that picture from above, you see this picture.
So that dotted line is the intersection of the ceiling, which is tipping like that,
and then the floor which is supporting the thing from below.
You can also see a cartoon: you just lift this and then that's the ceiling, pi prime,
and the floor is pi. And then the common cone,
tangent cone to these two spheres, is like that
and tip is lying in the intersecting line between the ceiling and the floor, okay.
So by lifting the, this complicated problem to the third dimension, the problem goes away.
In fact, if you look at it the right way, you know there used to be one of those
interesting pictures where you look at it with some out-of-focus sight,
suddenly it jumps out at you into 3D, and it's a bit like this.
If you look at it the right way, you can start seeing it like a three-dimensional picture
and then the theorem is clear. There is no problem.
I mean, of course,
these three points must be on the same line because that's how I see it.
[Brady:] This is actually a sequel to a previous video with the Professor about earthquakes and spheres,
hopefully you've caught that one too. Or if you just want to see more videos with the Professor,
click on the link that's on the screen or in the video description.
And, while you're in the video description,
you'll find details about these Numberphile playing cards which are coming soon!