Major and minor arcs, and why an arc's central angle is double what it subtends on the circle (Theorem 9)

0:00Mark two points on a circle. The run of circle between them is an arc, and the two points are its ends. But there are two runs, not one. Go round one way and you get a short arc; go round the other way and you get a long one. Both have exactly the same two ends. The longer one is called the major arc, the shorter one the minor arc.

0:24So naming an arc by its two ends alone does not say which of them you mean. The fix is a third letter, taken from the arc itself. Pick any point lying on the run you mean, and put its name in the middle. Arc A X B is the route through X. Arc A Y B is the route through Y. Same two ends, two different arcs, two different names.

0:50This is not fussiness about notation. Everything that follows is a statement about one arc and not the other, and a name that cannot tell them apart cannot state it. Drop the middle letter and the theorem stops being true half the time. So what angle does an arc make at the centre? Here is the definition that does all the work. Start a radius at one end of the arc.

1:18Turn it, keeping its moving end on the arc you chose, until it reaches the other end. The angle is how far it turned. Not the gap between the two radii at the finish. The turning. Two arcs, two different runs, two different amounts of turning. The wedge at the centre gets filled in by the sweep, so its colour tells you which arc produced it. And now something falls out immediately.

1:47Sweep along the minor arc and you turn through less than a straight angle. Sweep along the major arc, the long way round, and you turn through more. A major arc reads past a hundred and eighty degrees. That sounds wrong the first time. An angle at a centre, bigger than a straight angle? Under this definition, yes, and for a major arc it has to be. And the two sweeps always total a whole turn, because between them they cover the circle exactly once.

2:22So put a protractor on it. A reading under a hundred and eighty, and the route you traced was the minor arc. Over a hundred and eighty, and it was the major one. One number, one classification, and no eyeballing of lengths required. Here is a warning worth having. If you trace both arcs on one pair of ends and both come out under a hundred and eighty, you have traced the same route twice.

2:52Across four hundred and sixty two traced routes, two hundred and twenty came out over, twenty two landed exactly on the line, and two hundred and twenty came out under. Never two unders on the same pair of ends, because the two always add to a whole turn. There is a second place an arc makes an angle, and it is not the centre. Take a point of the circle that is not on the arc, and join it to the arc's two ends.

3:24That gives you an angle as well, taken out on the rim instead of at the middle. The condition matters. The point has to be off the arc. Put it on the arc and you are looking at a different picture, and what you measure is the other arc's business. So every arc carries two readings. One at the centre, and one from anywhere off it. The question is whether the two are related.

3:52Try it and see. Fix an arc, mark three points of the circle off it, and measure the angle from each of the three. Three viewing points, three separate measurements. They come out the same as each other, and each one is half the reading at the centre. That is worth being surprised by. Nothing in the drawing says the reading should hold steady as you slide the viewpoint round.

4:19But three points are three points. It tells you what to go and prove. It does not do the proving. So prove it. Call the centre C, and call the viewing point D. Join D to C, and carry the line on until it cuts the circle again, at a point we will call E. That line is a diameter, and it splits the picture into two halves. In the first case E lands on the arc itself, somewhere between its two ends.

4:50The angle at the centre is then cut into two pieces by that diameter, and so is the angle at D. If each piece at the centre is twice its matching piece at D, then adding them finishes the job. So the whole thing comes down to one piece. Look at one of them. The centre joins to one end of the arc and to D, and both of those are radii, so they are the same length.

5:20Two equal sides means two equal angles, at the two other corners. Now the diameter carries on past the centre, and the angle it makes on the far side is exterior to that triangle. An exterior angle equals the two interior angles it is not touching, added together. Those two are equal to each other, so the exterior angle is exactly twice either one of them. There is the doubling, and notice that it is not about circles at all. It is the exterior angle rule meeting a pair of equal radii.

5:59The other half of the picture does precisely the same thing, and the two halves add. Now move D, and run the same argument again. The diameter through D still cuts the circle at E, but this time E lands off the arc. The two doubled pieces are still there, obtained in exactly the same way as before. What changes is how they go together. The angle at the centre is now the difference of the two pieces rather than their sum, and the angle at D is the difference of its own two.

6:37Subtract instead of adding and the doubling survives anyway, because both terms carry the same factor of two. Same two triangles, same exterior angle, one sign changed. It is tempting to say there are two cases because there are two arcs. There are not. The arc was settled once, by the definition, and it never came back. The two cases are about where the diameter through the viewing point happens to re-cut the circle, which is a fact about the drawing rather than about the mathematics.

7:15And they are not even evenly matched. Across the configurations checked here, two thousand two hundred of them landed on the arc and one thousand nine hundred and eighty landed off it. There is a third landing that nobody mentions, four hundred and forty of them, where the diameter runs exactly to one end of the arc and one of the two triangles collapses to nothing at all. The result still holds there, but the argument as written does not reach it.

7:49Now use the thing. An arc making seventy degrees at the centre makes thirty five at every point of the circle off it. The other arc on the same two ends sweeps two hundred and ninety, and makes a hundred and forty five. Notice that half of a reflex angle is not reflex. A hundred and forty five is obtuse, and that is all it is. Notice as well that thirty five and a hundred and forty five add to a straight angle.

8:21That is not a coincidence, and it is completely invisible unless you let the major arc read past a hundred and eighty. One last thing. If the two ends are diametrically opposite, both arcs sweep a straight angle and every point of the circle reads a right angle - the single place where being off the arc stops mattering. Four thousand six hundred and twenty viewing points were checked. Off the arc, the doubling held at every one of them.