The points that are the same distance from two given points

0:00Here is a house. Not a real one — a drawing made with ruler and compasses, and the job is to make it again. Look at what you are given. Every line round the outside is five centimetres. The two walls, the floor between them, and the two slopes of the roof. All five. There is a door standing on the floor, one centimetre wide and two centimetres tall. And along the bottom of the roof there is no straight line. There is a curve, bulging gently downwards from the top of one wall to the top of the other.

0:38Five points to place. Four of them can go down straight away. The fifth cannot, and that is the whole point. Before drawing anything, work out the order. Call the bottom corners D and E, the tops of the walls B and C, and the point at the top of the roof A. Now ask, about each in turn, whether you could put it on the paper right now. D can go anywhere. Nothing is holding it.

1:08E is five centimetres from D, along the floor. B is five straight up from D, C five straight up from E. Also easy — a distance and a direction each. And A? A is five from B. But in which direction? Five from C as well — again, which direction? Neither tells you which way to point the ruler. A distance with no direction is not somewhere you can put a pencil.

1:38The walls go down first, because nothing about them waits on anything else. Draw the floor, five centimetres, from D across to E. At D, turn a right angle and go five up. That is B. At E, do the same. That is C. Three sides of a square, with the fourth side missing. Now the door. One centimetre across, two up, standing on the floor. And notice: you are told how wide the door is and how tall, but not where along the floor it stands.

2:11That is not an oversight. Slide it left or right and it is still the door you were asked for. Part of reading a problem is noticing which questions it declines to answer. Now A. Two things are true about it: A is five centimetres from B, and A is five centimetres from C. Take those one at a time, and watch how little each manages alone. Suppose all you knew was that A is the same distance from B as from C. Where could it be?

2:45Straight up from the middle, certainly. But also higher. Also lower. Also below the floor. Every point on the line running up through the middle of B and C is the same distance from both — endlessly many of them. So being equally far from two points puts you nowhere in particular. It hands you a whole line. What narrows it down is the other half of the information. Not just equal. Five.

3:16There is a way to attack that with a ruler, worth doing badly on purpose. Put the end of the ruler on B and swing it, watching the five centimetre mark. Every position hands you a point five from B. So far so good. Now check the other condition. How far is that point from C? Six point two. Swing a bit. Four point four. Swing back. Five point three.

3:43You are hunting. You can go on hunting, and you will get close. But whatever you land on, you landed on it by luck — and tomorrow you would land somewhere slightly different. A construction should not depend on how patient you are. So stop chasing one point, and draw all of them instead. Open the compass to five centimetres. Put the point on B, and go the whole way round.

4:12Every single place on that circle is exactly five from B. That is the first condition drawn out in full — nothing left out, nothing guessed. And it is enormous — more points than anybody could ever check one by one. Which sounds like the opposite of progress. It is not. Because there is a second condition still to come, and its whole job is to throw nearly all of them away.

4:42Same compass. Same five centimetres. Different centre. Put the point on C this time, and go round again. Now there are two circles on the paper, and they overlap a great deal. Look at what each is saying. Everything on the first is five from B. Everything on the second, five from C. And A has to be on both. So A is not merely somewhere on a circle any more. It is in the small handful of places the two circles share.

5:15And you can see them. They are exactly where the two curves cut across each other. There are two of them. Not one. Two. Worth stopping on: it is easy to draw two circles and expect a single answer. One is up above the walls, where the top of a roof belongs. The other is down between them, close to the floor. Both are five centimetres from B. Both are five from C.

5:44Which one you want is settled by the picture, not by the numbers. You are drawing a house, so you take the upper one. Mark it A. And there is a reason there were two. B and C are five apart and the radius is five, so each circle runs through the other's centre. Two circles overlapping like that always cross on both sides of the line joining their middles.

6:12Look back at those two full circles and ask how much of that ink did any work. Almost none. Everything that mattered happened in one small region above the walls. The rest swept round the back and came home again. So do it again with less. From B, draw only a short stroke where you expect the crossing to be. Then from C, another short stroke across it. They meet in exactly the same place as before.

6:43A stroke reaching twenty degrees either side of the answer is plenty — under a ninth of the circle. The arcs are not a rough version of the circles. They are the same construction with the unused part left undrawn. Join A to B, and A to C. The two slopes of the roof are done. Now the curve along the bottom. Its centre is A, the point you have just found.

7:11Open the compass to five once more, put the point on A, and draw from B to C. And here is the neat part. A is five from B and five from C, because that is how it was chosen. So the arc runs through both of them without your having to aim. How much of a circle is it? Not a half. Nowhere near. A is five from B, five from C, and B and C are five apart. Three equal distances — so the arc turns through sixty degrees, a sixth of a circle.

7:47Which is why the roof is a shallow curve and not a dome. That move — one arc for each of two known points — is the whole technique. Build the same house bigger. Every line seven centimetres instead of five. The walls change. The compass setting changes. The method does not. The top comes out about thirteen centimetres up rather than nine, and the angle there is still sixty. Same shape, larger.

8:19Any time a point has to sit a known distance from one known point and a known distance from another, this is how you find it. One arc for each condition. The point is where they cross. And there will be two crossings, and you will have to decide which you meant. One last thing, and it uses the same move to break a belief. Draw two lines five centimetres long out of one corner — deliberately not at a right angle. Sixty degrees, say.

8:52Now close the shape up, so all four sides are five. The fourth corner has to be five from one loose end and five from the other. Two arcs. And they cross twice, as always. One crossing is the corner you started from — it is five from both, so the arithmetic hands it back. Ignore it, take the other. Four sides, every one five centimetres, and not a right angle anywhere.

9:21So a four-sided shape with four equal sides does not have to be a square. The two lines across the middle come out at five and about eight and a half — where a square's would have matched.