Chords of equal length cut off equal central angles, and the converse (Theorems 2–3)

0:00Pull a thread tight across a wheel, from one point of the rim to another. That is a chord. A segment with both of its ends on the circle. Now lift it off and lay it down somewhere else on the same rim, without changing its length. It looks like the same chord in a new place. What we want to know is whether it behaves like the same chord.

0:24Because the two ends and the hub make an angle at the centre. Does that angle come along for the ride? If it does, then a chord's length and the angle it makes at the centre are two names for one piece of information. You could hand over either and get the other back. That is a strong claim, and turning a wheel is not going to earn it. Start by noticing what a chord brings with it.

0:53Take any chord at all. Join each of its ends to the centre. Those two joins are radii, so they are equal - not by luck, but because that is what a radius is. So every chord comes with an isosceles triangle already attached. And that is the whole reason this topic works out the way it does. Two of the three sides are pinned at the radius, whatever chord you happened to pick.

1:21Which leaves exactly one thing free: the third side, or what comes to the same thing, the angle between the two pinned ones. Fix either and the other has nowhere left to go. That is a promise rather than a proof, so here is what a picture can do and what it cannot. Take one chord with its two radii, and turn the whole shape about the centre. Turn it once. Turn it again. The same triangle, in three places.

1:53Every point stays the same distance from the centre, so the shape lands back on the circle each time. Length survives the turn. The angle survives it. Position does not, and position was never the thing we cared about. That is genuinely persuasive, and it is not enough. It covers chords you obtained by turning one chord. Two chords that merely happen to be the same length, arrived at independently, are not covered at all. Those need an argument.

2:29So here is the claim, one way round. Take two chords of one circle with the same length. Anywhere on it. Not related to each other by anything. Then the angles they make at the centre are equal. To argue it, look at the two triangles - each chord with its own two radii. In the first, two sides are radii. In the second, two sides are radii. One circle, so all four of those are the same length.

3:00And the third sides, the chords themselves, are equal - because that was the thing we were given. Three sides of one triangle match three sides of the other. That is the side-side-side test, and it says the two triangles are congruent. Congruent means every matching part agrees, and we get to choose which part to read off. Read off the angle at the centre. In the first triangle that is the angle between the two radii. In the second, the same.

3:35They correspond, so they are equal, and that is exactly the claim. Notice what did the work there. Not the drawing. The two radii, which were equal before anybody chose a chord at all. Without them there would be one matching pair of sides and no test would apply to anything. Say it without the circle and it is a statement about isosceles triangles: two of them with the same equal sides and the same base are congruent.

4:09Now turn the question round, which is a different question and not a restatement of the first. This time you are handed the angles instead. Stand at the centre with your arms held at a fixed opening. Point them at two points of the rim. Now swing round, keeping that opening exactly, and point somewhere else entirely. The claim is that the chord across your fingertips has not changed its length.

4:39The arms arrange the claim so that you believe it. They also assume, quietly, that the swing lands exactly where it needs to. That assumption is the thing being proved. So argue it properly instead. The same two triangles. Each chord with its two radii. But what is given has changed. Two sides in each triangle are radii, exactly as before. The third side is not given this time. The angle between the two radii is.

5:14Two sides, and the angle sitting between them. That is the side-angle-side test, and it gives congruence too. So read off the third side, which is the chord, and the two chords are equal. Same picture, different given, different test. Which raises the obvious question. Why not use the same test both times? Because the test you may use is decided by what you were handed, not by what you would prefer.

5:46The first time we were handed three sides. Only the three-sides test fits three sides. The second time we were handed two sides and the angle between them. Only side-angle-side fits that. And you cannot help yourself to any two sides and any angle. Two sides and an angle that is not between them do not pin a triangle down at all. In a small grid of points, twenty one different bundles of exactly that kind each admit two genuinely different triangles.

6:20So the third side is not determined and no theorem follows. Which tool is available really is decided by the data. Both directions together say something worth stating plainly. On one circle, chord length and central angle carry exactly the same information. That was checked rather than trusted. Fifty three points on a circle of radius five, every pair of them joined: one thousand three hundred and seventy eight chords.

6:55Each chord's length measured from its own two ends. Each chord's angle measured at the centre. Neither of those calculations ever looked at the other. Three hundred and forty seven different lengths turned up. Not one length was ever seen with two different angles, and no two different lengths ever shared an angle. Three thousand six hundred and seventy nine pairs of equal chords, and every single pair made congruent triangles at the centre.

7:28The match is not only one-to-one. It runs in one direction only. Sort those three hundred and forty seven lengths from shortest to longest, and watch the cosine of the angle at the centre. It falls at every single step. Three hundred and forty six steps, with no exceptions anywhere. Longer chord, wider angle. Always, and never the other way. Which is what makes the correspondence usable. Give a length and there is one angle to hand back, and it is easy to say which one.

8:05But usable is not the same thing as proportional. And that is where the idea usually comes unstuck. Take a circle of radius ten. A chord as long as the radius - ten units - makes a triangle with three equal sides. All three are ten, so that triangle is equilateral, and the angle at the centre is the angle an equilateral triangle has. Now double that angle. The cosine at the centre goes from a half to minus a half, and the chord squared goes from a hundred to three hundred.

8:44So the chord goes from ten to a little over seventeen. It does not go to twenty. And that is not a quirk of this example. Double any central angle at all and the chord grows by less than double. It creeps towards doubling as the angle you started from shrinks away to nothing, and it never once arrives. One last thing, and it is easily missed. All of this lives inside one circle.

9:15A chord six units long makes one angle on a circle of radius four, and a different angle on a circle of radius five, and a different one again on every radius up to twelve. Nine radii, nine different angles, from the one chord length. The chord alone does not settle the angle. The chord and this circle do. With that said, here is what the pair of theorems actually buys you.

9:43You may pick up a chord, carry it anywhere you like on its own circle, and treat what you get as the same chord. Not because it looks the same. Because the two radii it arrives with leave it no choice.