Off the circle the angle changes: points inside and outside compared (Fig. 5.25)

0:00Here is a chord, and here is a point looking at it. The point reads an angle - the angle between its two lines of sight, one to each end of the chord. Now move the point somewhere else. Does the reading change? There are two answers a reasonable person might defend. One: the chord fixes the angle, so wherever you stand you read the same thing. Two: the reading depends on where you stand, and the one place it stays put is the circle.

0:32Those cannot both be true, and the difference between them is not a detail. If the first were right, an angle would tell you nothing at all about where you were. So set it up as an experiment. One circle. One chord across it, with its two ends marked A and B. Seven places to stand, and all seven on the same side of the chord - the side matters, and at the end you will see exactly how much.

1:03Three of the seven are inside the circle. Two of them are on it. Two are outside. From each place, look at A, look at B, and write down the angle between those two lines of sight. Seven readings. That is the whole experiment. Start inside. Three points, three readings - and the three readings are all different from each other. So the first answer is already dead. A chord does not look the same from everywhere.

1:36But notice what the three do share. Every one of them is wider than what we are about to read on the circle itself. Walk right in to the centre and the reading there is exactly double the circle's. And the centre is not even the extreme case. Slide back out towards the chord and the two lines of sight open further still, towards a straight line. Now stand on the circle.

2:04Two places, well apart, on the same arc. Same reading. Not close - the same. That is the constancy we already know about, and it is worth saying precisely what it claims. Every point of one arc sees the chord at one angle, and that angle belongs to the arc rather than to any point of it. Step outside the circle. Two more places, two more readings - different from each other, as they were inside.

2:36And both of them narrower than the circle's. So the seven sort themselves out cleanly. Three wide readings inside, one shared reading on the circle, two narrow readings outside. And the widest reading anyone got from outside is still narrower than the tightest reading anyone got from inside. The three groups do not overlap. Now be careful, because seven points is a demonstration and a demonstration proves less than it looks like it proves.

3:11What it shows is that the readings differ. It does not show that they have to differ that way round. Those are two separate claims, and the second one is much the stronger. An experiment that came out this way seven times could still come out the other way on the eighth. So try more of them. Five hundred and eighty-nine places were tried - inside, on the circle and outside - and not one of them broke the pattern.

3:41Which is reassuring, and still not a proof, because five hundred and eighty-nine is not all of them either. So the direction is not something to be observed. It has to be argued. Here is the argument, and it needs exactly one fact about triangles. Take any triangle and carry one of its sides straight on past a corner. The angle you open up outside is the other two corners' angles added together.

4:13Added - so it is strictly bigger than either of them alone. Now stand outside the circle and look at A. Your line of sight crosses the circle on the way, at a point call it E. Then you, E and B make a small triangle, and the angle E reads is one of those carried-on angles. So E's reading beats yours, and E is on the circle. Stand inside instead, on the line between A and E, and the whole picture flips: now your angle is the carried-on one, and you beat E.

4:48One fact, both directions, and neither of them depends on where the points happened to be put. You can also just walk it. Stand just under the chord, between its two ends, where your lines of sight are almost a straight line. Now walk straight away from the chord. The reading falls. It keeps falling. It never once turns back on itself. And somewhere on the way out it passes the circle's value - exactly once, and at the exact step where you cross the circle.

5:23Not before, not after, and never twice. That is the whole of this topic in one movement. Put numbers on it. Say this chord cuts a hundred degrees off the circle, so every point of the far arc reads fifty. Take two chords that cross inside, at some interior point. The angle there is one pair's reading plus the other pair's. If the second pair cuts off thirty-six degrees - that is eighteen - the interior point reads sixty-eight.

5:57Now carry those same two chords on until they meet outside instead. Same step, one sign different: minus, not plus. Cut off thirty-two and you subtract sixteen, and the exterior point reads thirty-four. Sixty-eight inside, fifty on, thirty-four outside - and the plus and the minus are the whole explanation. Now turn the whole thing round, because backwards is where it earns its keep. The circle's reading for this chord is fifty.

6:32Someone is standing where you cannot see them, on the side you know, and they report the angle they read. They say sixty-one. Sixty-one is more than fifty, so they are inside. They say forty-one. That is less, so they are outside. They say fifty, and they are on the circle. You have placed a point without measuring one single distance. The angle gave it away - and it could only do that because the angle is not the same everywhere.

7:05Which is the point of the whole video. A test that everything passes is not a test. If a chord really did look the same from every point of the plane, then equal angles would be telling you nothing, and there would be nothing to build on top of it. The constancy is only useful because it fails everywhere else. And there is a nicer way to see the same thing.

7:31Every reading picks out an arc: the points that share a reading are never scattered about, they always lie together on one arc through A and B. There is a whole family of these arcs, nested, one for each reading. The circle we started with is simply the arc whose reading we happened to name first. One last thing, and it is the condition that has been sitting there since the second sentence.

8:00The same side. Take a chord and put two points opposite each other, mirror images across it. By symmetry they read exactly the same angle. Equal angles - so are they on one circle with A and B? Almost never. Run through every height you like and there is exactly one that works: the one where both readings are right angles, and the chord is the full width of the circle.

8:30Every other mirror pair reads equal angles and is not on one circle at all. So equal angles do not by themselves put points on a circle. Equal angles on the same side do. The condition is not a formality anyone forgot to delete - drop it and the statement is false.