None, one or two, depending on where the point sits

0:00Every question so far has started with the circle. Here is the same subject asked the other way round. Fix a point first. Anywhere you like on the page. Then ask how many tangents to this circle pass through that point. The answer is none, or one, or two. Never anything else. And which of the three you get has nothing to do with the tangent, and nothing to do with how big the circle is. It turns on one thing only. Is your point inside the circle, on it, or outside it?

0:37That is worth saying slowly, because it is a strange kind of answer. The count is a fact about the POINT. So let us take the three cases in order, and in each one ask not only what happens, but why nothing else can. Case one. Put the point inside the circle, and call it P. Draw a line through P. Any line at all. It comes in from outside, and it has to reach P, which is inside, so somewhere along the way it crossed the circle. Keep following it and it has to get out again, so somewhere it crosses a second time.

1:17Two crossings. That line is a secant. Turn it a little and try another direction. Two crossings. Another. Two crossings again. Every direction, the same answer. From a point inside the circle there is no tangent at all, and the count is nought. The obvious objection is that we have not tried hard enough. Somewhere between those directions there must be one that only grazes. There is not, and the reason takes one line.

1:52Take any line through P, and find the point of it nearest the centre. The segment from the centre down to that point is the shortest reach from the centre to that line. Now P is itself a point of the line. So the shortest reach is at most as long as the segment from the centre to P, because nothing can be shorter than the shortest. And P is inside the circle, so that segment is shorter than the radius.

2:22Put those together. Every line through P comes closer to the centre than the radius does. And a line that comes closer than the radius has to cut the circle. Twice. There is nothing there to aim at. The failure was never about aim. Case two. Put the point on the circle itself. Now there is exactly one tangent there, and that word exactly is quietly doing two separate jobs.

2:53One job is to say there is AT LEAST one, that a tangent through this point exists at all. You settle that by producing one. Draw the radius out to the point and put a line up square to it. The other job is to say there is AT MOST one, and that comes from somewhere else entirely. Any tangent touching here has to be square to the radius at this point, and through a given point of a given line there is only one line square to it.

3:25Existence and uniqueness. Two different statements from two different places, and they are worth keeping apart. A drawing of one tangent shows you the first and tells you nothing whatever about the second. Case three. Put the point outside the circle. Now two tangents appear, one on each side, touching the circle at two different points. Watch what happens as the point moves. Bring it in towards the circle and the two tangents swing together, and the two touching points slide towards each other.

4:03The moment the point arrives on the circle they have merged, and the two have become the single tangent of case two. Push the point back out and they separate again. Take it a long way off and the two tangents become almost parallel. Almost, but never quite, and never fewer than two. Two is easy enough to see. Why can there never be three? Here is an argument that closes it off. Suppose a tangent from our outside point P touches the circle at T.

4:38We know the radius drawn to T meets that tangent at a right angle. So in the triangle O T P the corner at T is square. Now find the midpoint of the segment from the centre O to the point P, and draw the circle that has that segment as its diameter. Every touching point has to sit on that new circle. You can check it as a plain distance. T is exactly as far from the midpoint as O and P are, and that is what the right angle at T buys you.

5:13So the touching points are not free to be anywhere. They have to be points shared by two circles, the original one and this new one. And two different circles share at most two points. There is no room for a third. One thing is worth noticing before we go on, because it says what the count is really about. Take the circle away and put an ellipse there instead. Squash it, stretch it, move it off to one side.

5:44Ask exactly the same question. A point inside gives no tangents. A point on it gives exactly one. A point outside gives exactly two. The same three answers, for the same three reasons. The count was never about the shape being round. Hold on to that, because everything from here on does use roundness, and on an ellipse it falls apart. Two tangents from an outside point of a circle turn out to be the same length as each other. Two tangents from an outside point of an ellipse are not.

6:24Which brings us to the thing case three makes possible and the other two do not. A tangent is a LINE. It runs on for ever in both directions. It has no length, and asking how long it is makes no sense. But from an outside point P, touching at T, there is now a piece of it with two ends. From P to T. That piece has a length, and it is called the length of the tangent from P.

6:53Read the phrase carefully. It is not the length of the tangent, because a line has none. It is the length of the tangent FROM a stated point, and it changes the moment you move the point. Case one has no such segment, because there is no tangent to cut a piece from. Case two has one, but the point and the touching point are the same place, so the length is nothing.

7:22Now the useful part. You can find that length without drawing anything at all. Three things sit in one right triangle. The centre O, the touching point T, and the outside point P. O T is a radius, so its length is r. O P is the distance from the point to the centre, and I will call it d. And P T is the length we are after. The angle at T is a right angle, so O P is the hypotenuse. It is the side facing the right angle.

7:57Pythagoras then says d squared is r squared plus the tangent length squared. So the tangent length squared is d squared MINUS r squared. Radius five, point thirteen from the centre. A hundred and sixty nine take away twenty five is a hundred and forty four, so the tangent is twelve. Turn it round. Tangent twenty four long, point twenty five from the centre. Six hundred and twenty five take away five hundred and seventy six is forty nine, so the radius is seven.

8:34Or a point five from the centre with a tangent four long. Twenty five take away sixteen is nine, so the radius is three. There are two things this length gets confused with, and numbers settle both. It is not the distance from the point to the circle. Take that last case but one. Radius seven, point twenty five from the centre. The nearest part of the circle is twenty five take away seven, which is eighteen away. The tangent is twenty four long. Those are not the same number and they are not even close.

9:14The reason is that the tangent does not run straight in at the circle. It runs off sideways and touches it further round. And it is not the distance to the centre either. That was twenty five. The tangent is shorter than that, always, because we subtracted the radius squared to get it. So the tangent length sits strictly between the two. Longer than the gap across to the circle, shorter than the reach to the centre. Every time.

9:48Here is the thing I like most about this topic. That one expression, d squared minus r squared, already knows all three cases. Keep the radius at five and slide the point. Put it three from the centre. Nine take away twenty five is minus sixteen. A negative number has no square root, so there is no length, because there is no tangent. That is case one. Put it five from the centre, so it lands on the circle. Twenty five take away twenty five is nothing. The square root of nothing is nothing, so the length is nothing, which is exactly right. The point and the touching point are the same place. Case two.

10:35Put it thirteen out. A hundred and sixty nine take away twenty five is a hundred and forty four, whose root is twelve. Two tangents, each of them twelve long. Case three. One expression, three verdicts. The arithmetic tells you which case you are in before you have drawn a thing. One last thing, and I am going to leave it open on purpose. Go back to the outside point with its two tangents, and measure both pieces. From the point to the first touching point, and from the point to the second.

11:14They come out the same length. Try another point outside. Same again. Change the radius, move the circle: still the same. The arithmetic we just did already hints at why, because the expression only knows d and r, and both tangents share the same d and the same r. But a hint is not a proof, and seeing a pattern is not showing it must hold. This one has a proper proof waiting, and it is where we go next.