Three points not in a line: exactly one circle (Theorem 1)

0:00Two given points did not pin a circle down. They left a whole line of possible centres, and a circle for every point on it. Now add a third point and ask again. How many circles pass through all three? The answer comes with a condition attached, and the condition is where the interest is. If the three points sit along one straight line, the answer is none. Not one. None at all.

0:29And if they do not, the answer is exactly one. Exactly one is a strong thing to say. A circle exists, and there is no second one anywhere. Two claims, and they come out of a single picture. Take the failure first, because it is the half that gets skipped. Three points along a straight line. Call them A, B and C. A centre serving all three has to stand as far from A as from B, and as far from B as from C.

1:02Each demand is a line of possible positions. The perpendicular bisector of that pair. Draw both. They come out parallel. Same direction, different positions, never meeting. No point in the plane meets both demands, so there is no centre, and so no circle. A search over three thousand seven hundred and twenty one positions finds nowhere at all. That is what nothing looks like when you check. Parallel, though. Why? A picture that happens is not a reason.

1:39Both bisectors stand at right angles to the line the three points sit on. Two lines at right angles to the same line point the same way. That is the whole of it. And they are not the same line, because they cut across at different places. Which rules out the tempting explanation. The failure is not that the circle would be too big to draw. That is a picture again. The failure is that two demands have no common solution. Size has nothing to do with it.

2:15There is another way to put the same thing, different enough to be worth a minute. A straight line cannot cut a circle in three places. At most two. Those are the same statement. Three points of a circle standing in a line would be a line cutting it three times. Here is a circle of radius five on a grid. It carries twelve points with whole number coordinates. Test every triple of the twelve for straightness. Not one is in a line.

2:47Take every pair instead and count how many of the twelve sit on the line through them. The most any pair gets is two. A circle bends away from every line it touches. Three shared points would need it to stop bending. So to the other case. Three points not in a line. The claim is that there is exactly one circle through them. Split that in two, because it is two statements wearing one sentence.

3:17Existence: at least one such circle can be produced. Uniqueness: there is no second. Normally those take separate arguments, the second usually harder. Here they do not. Both fall out of one small fact about lines. Two lines that are not parallel cross at exactly one point. Not at least one. Not at most one. Exactly one. Set it up. Three points, A, B and C, not in a line.

3:49A centre has to stand equally far from A and from B. That is one line of positions. It also has to stand equally far from A and from C. That is a second line. Two demands, two lines, one plane. This time the lines are not parallel, and so they cross. Mark the crossing. Call it O. O sits on the first line, so it is equally far from A and B. It sits on the second, so it is equally far from A and C.

4:22Which makes all three distances the same - the only thing a centre was ever asked to do. Existence first. Compass point at O, open out to A, and draw. B lands on the circle, because it stands at the same distance. So does C. There is the circle. We built it, rather than assuming it and hunting for its middle. Now uniqueness, out of the same picture, nothing new added.

4:52The centre of any circle through all three meets both demands, so it sits on both lines. The lines cross once. So there is one candidate, and it is O. And once the centre is settled the radius is settled too. It is whatever reaches A. One centre, one radius, one circle. The same crossing, read a second time. So what separates the two cases? Solve the two demands together as a pair of equations and a single number decides everything. The determinant.

5:30When it is not zero there is one solution. When it is zero there is none. And that number is not merely connected to the points being in a line. It is four times the area of the triangle they make. No area means in a line. So it vanishes exactly when the points are collinear and never otherwise, checked on every triple a seven by seven grid can make.

5:57Watch it fail gradually rather than suddenly. Sink the third point towards the line in forty steps. The radius grows at every step. By the fortieth the radius squared is past four hundred. And when the point lands on the line, there is no crossing at all. Three points not in a line make a triangle, so the circle picks up a name that mentions one. The centre is the circumcentre. The circle is the circumcircle.

6:30The circle circumscribes the triangle. The triangle is inscribed in the circle. Four words for one drawing, each naming a different part of it. Notice what the circumcentre is defined by. Not by looking central. By standing equally far from the three corners. Those are not the same thing, and the difference is about to matter. One more thing. We used two bisectors; there is a third, for the pair we left out.

7:01It runs through the crossing every time, on all seventeen thousand six hundred triangles tested, and it makes no difference which two you pick. So where does the circumcentre actually sit? Everybody's mental picture puts it inside the triangle. Sometimes it is. Take every triangle a seven by seven grid can make. Seventeen thousand six hundred of them. Sort each by its largest angle and record where the centre falls.

7:35Three combinations come back, and only three. Acute: the centre is inside. Obtuse: the centre is outside, past an edge entirely. Right angled: the centre is on the boundary. Nothing else ever happens. And the counts hold a surprise. Three thousand eight hundred and eighty acute, ten thousand seven hundred and sixty obtuse, two thousand nine hundred and sixty right angled. The acute case, the one every drawing of this uses, is the rarest of the three.

8:13The right angled case is sharper than just on the boundary. The centre sits at the midpoint of the longest side, the one facing the right angle. Every single time. Which means the radius is exactly half of that side. It is also the same fact as a statement about angles that turns up wherever a right angle meets a circle. And you can predict the position from the angles alone, before drawing anything.

8:44Two angles of seventy and sixty leave fifty for the third. All three under ninety, so the centre falls inside. One angle of a hundred leaves eighty to share, so neither of the others reaches ninety. The hundred is the largest, it is past ninety, and the centre falls outside. The largest angle decides it. Build one and check it by hand. Sides of six, seven and seven. Bisect two of the sides, mark the crossing, then measure out to each corner in turn.

9:22The three lengths come out equal. Not luck with the ruler: they are three radii of one circle. Worked out exactly, that radius squared is two thousand four hundred and one over one hundred and sixty, which puts the radius between three point eight seven and three point eight eight. One last question, to stop exactly one becoming a reflex. How many triangles can one circle hold? Turn the triangle about the centre. Every corner stays on the circle, because turning about the centre leaves every distance from the centre alone.

10:03Thirty three different turns give thirty three different triangles, all with the same three side lengths, all on that one circle. So a triangle has exactly one circumcircle. A circle does not have exactly one triangle.