Two points: infinitely many circles, centres on the perpendicular bisector

0:00Here is a question that sounds as though it has a one word answer. How many circles pass through two given points? Most people say one, and they say it because of something they already know. Through two points there is exactly one straight line. Try every line the plane can name and exactly one of them goes through both. So the instinct is to expect the same tidiness from circles.

0:28It is the wrong instinct, and how wrong it is turns out to be the interesting part. Two points do not pin a circle down. They do not pin it down a little bit either. They leave a whole line's worth of them. The move that answers this is to stop asking about circles. A circle is fixed the moment you know its centre and its radius. But if it has to pass through a point you have already named, then the radius is not free. It is whatever reaches that point.

1:03So the circle is fixed by its centre alone. Which turns counting circles into counting centres, and that is a much easier thing to count. And a centre that works has exactly one job to do. It has to stand the same distance from the first point as from the second. That is the whole condition. Nothing else is required of it, and nothing else is allowed to matter. So take two points six units apart, and go looking for centres.

1:36Take every point of a grid, four hundred and forty one of them. Give each one the radius that reaches the first point, then ask whether that same radius also reaches the second. Most of them fail. Twenty one survive. And the very first one anybody finds is the obvious one: the middle of the segment joining the two points. Three units from one end, three units from the other.

2:05Draw its circle, and both of the given points are sitting on it. That is one circle through two given points, found with no theory at all. That circle is worth a second look, because it is special twice over. The two points sit at opposite ends of it, with the centre exactly between them. So the segment joining them runs straight through the centre. It is a diameter. And it is the smallest circle there is through the pair.

2:37No centre anywhere gives a radius shorter than three, and three is half of six. Half the distance between the two given points. So the least radius available is always half the gap, whatever the gap happens to be. Which answers a question people find surprisingly hard: what is the smallest circle through two given points. Now. Are there others? Look at where those twenty one centres actually sit. Every single one of them is at the same place along the segment. Halfway.

3:14Above it, below it, close in, far out, but always halfway along. They form a straight line, and it is the line that cuts the segment in half at right angles. Its name is the perpendicular bisector. And that is not a coincidence arriving from somewhere else. It is exactly the set of points standing equally far from the two ends, which is precisely the job we said a centre had to do.

3:45Here is the part it is tempting to skip. Showing that every point on that line stands equally far from both ends is only half of the claim. Watch what that half actually buys you. Take a short stretch of the line. Seven points. That stretch has a hundred and twenty eight subsets, counting the empty one and the whole thing. Run the equal distance test on every subset, and all hundred and twenty eight of them pass. Every last one.

4:17A test that everything passes cannot tell you which set you have found. The half that does the work is the other one. Nothing off the line is equally far from both. Step off the line, and the two distances squared come out at twenty nine and five. Draw the whole thing and it wants to be two pictures, not one. On one side, the phenomenon: two points with circles crowding through them, and no explanation anywhere in sight.

4:49On the other side, the explanation. The bisector drawn in as a dashed line, three of its points picked out and lettered, and the circle belonging to each of them. Two of those centres above the segment, one below it. Same two points every time. Different heights on the line. Different circles. Those are not two pictures of the same thing. The first one shows you that it happens. The second shows you why.

5:20So walk a centre up the line, and watch the radius. At the segment itself, at height nothing, the radius is three. Go up four and the radius is five. Go up twelve and the radius squared is a hundred and fifty three, which makes the radius about twelve point three seven. Notice what that is not. From height four to height twelve, the height trebled. The radius went from five to twelve point three seven, which is nowhere near three times five.

5:53Radius and height do not march in step, because the radius is the long side of a right angled triangle whose other two sides are the height and the three. Keep walking and two things happen at once. The radius grows past any number you care to name, so there is no largest circle through two points. There is no last point on a line to stand on. And the arc joining the two given points gets flatter.

6:25There is an exact way to say how much flatter. Take how far the arc bulges past the segment, and multiply it by the radius plus the height. You get nine. The half gap squared. The same number at every height, all the way out. So the bulge is nine divided by something that grows without limit. It shrinks towards nothing, and it never once reaches it. Almost straight is not straight.

6:55Which radii can you actually have? Everything from three upwards, and nothing at all below it. And there is a catch inside that worth knowing. Twenty one centres gave twenty one different circles, but only eleven different radii. Because a centre above the segment and its mirror image below stand the same distance from both points. Same radius. Different circle. So if somebody hands you two points and a radius and asks how many circles fit, the answer is two, and not one.

7:31Unless the radius is exactly the smallest, where the two mirror images are the same point and there is only one. Below that, none at all. Now count. One line of centres. One circle for every point on it. The points of a line cannot be counted, so neither can the circles. Widen the search from twenty one candidates to a hundred and one, and then to four hundred and one, and the number simply keeps climbing.

8:03It never stops. Infinitely many circles pass through two given points. But do not read infinitely many as anything goes. Every one of them has its centre on one particular line, and not one of them has a radius below half the gap. An infinite family can be held extremely tightly. One last thing, to stop infinitely many becoming a reflex. Ask the same question about squares. Two given points, and this time they have to be corners.

8:39Search for every four cornered figure with four equal sides and the right diagonals, and exactly three of them come back. The pair can be one side, with a square built on either side of it. That is two. Or the pair can be a diagonal, which gives one more. Three. Not infinitely many. Three. Now loosen it. Let the two points be merely somewhere on the boundary, rather than at corners, and the count runs away again.

9:11So the answer was never about having two points. It was about how tightly the shape was pinned.