A tangent as what a secant becomes when its two crossings merge

0:00Last time we sorted every line in the plane into three kinds, and we did it by counting. Share no point with the circle and the line is non-intersecting. Share two and it is a secant. Share exactly one and it is a tangent. That is a complete list. Nothing else can happen. Now we single out the third one, because it is the case everything after this is built on.

0:29Here is the trouble with defining a tangent as the line that shares one point. It is correct, it is checkable, and it tells you nothing about where such a line comes from. It makes the tangent sound like a separate species. A secant cuts, a tangent touches, and never the two shall meet. They are not separate at all. They are the same object at two settings of one dial, and the dial is the length of the chord.

1:00A secant cuts a chord. Slide it, turn it, do anything you like that makes that chord shorter, and when the chord reaches nothing you are looking at a tangent. So the question is not what a tangent is. It is what a secant becomes. Take a circular wire lying flat on a table, and a straight wire laid across it. Pin the straight one to the circular one at a single point, so it can swing about that pin but cannot leave it.

1:31The pin is on the circle, so that point is shared no matter how you turn the wire. It is one of the two crossings, permanently. The other crossing is free to move. Watch it. Turn the wire, and the second crossing travels round the circle towards the pin. Here it is at three positions, and I am going to measure the chord each time. That is the distance from the pin to the second crossing.

2:01At the first position the chord squared is nearly twelve. At the second it is three point two. At the third it is about a tenth. Twelve, then three, then a tenth. It is not just getting smaller, it is getting smaller strictly, every single step. And one more turn brings the second crossing all the way in. It arrives at the pin. The chord is now zero, and the two crossings are not close together. They are the same point.

2:34One shared point. That line is a tangent, and we did not define it that way. We measured our way to it. Now do it again from the other direction of turn. The second crossing comes round the other way, three more positions, and the chord squared falls from about five, to two, to a tenth again. It closes on exactly the same line. The two approaches meet. That matters, and here is why. If you only ever watched one side, you would think the tangent was where the turning stops. An endpoint.

3:12It is not an endpoint. Keep turning past it and the second crossing reappears, on the other side of the pin, and the line is a secant again. Over the whole fan of positions I tried, the far crossing sits on one side of the pin twenty five times, on the other side fifty six times, and on neither exactly once. Exactly once. The tangent is a position you pass through, not a wall you stop at.

3:45There is a second experiment, and it gets to the same place without turning anything. Draw a circle. Draw one line across it, a secant. Now draw a whole family of lines parallel to that one, on both sides. Every member of the family points in the same direction. The direction is fixed. The only thing telling them apart is how far each one is from the centre. So slide across the family and watch the chord.

4:16Wide in the middle, narrower as you go out, and at some point it stops existing altogether. That is the same story as the turning wire, told without any turning. The chord is the quantity, and its running out is the event. Let us get the chord exactly, because this is where the picture turns into arithmetic. Call the radius r, and call the perpendicular distance from the centre to the line d.

4:47Drop that perpendicular. Its foot is the middle of the chord, and half the chord, the foot, and the centre make a right triangle with the radius as its longest side. So half the chord squared is r squared less d squared, and the whole chord is twice the square root of that. Put in a radius of five. At the centre, d is nothing and the chord squared is a hundred, so the chord is ten, which is the diameter.

5:18Three fifths of the way out, chord squared sixty four, chord eight. Four fifths of the way out, chord squared thirty six, chord six. And at the edge, where d has reached five, r squared less d squared is nothing, and the chord is gone. I checked that against the drawing rather than against the formula, on every member of the family whose crossings land on exact points. The measured chord and the formula agree every time.

5:51Now the question the family answers that the turning wire does not. How many members of that family are tangents? Two. Exactly two, and you can see why from the chord. It runs out when d equals r, and the family passes the centre on both sides, so d reaches r once on each side. I ran that sweep over two hundred and sixty one parallel lines at six different radii, and the answer was two every time.

6:23Not because the routine is fond of the number two. Point it at a radius whose touching lines the sweep cannot land on exactly, and it reports none at all. And the two touching points are worth a look. They sit opposite each other through the centre, and the line joining them runs square to the whole family. So the two tangents are drawn at the two ends of one diameter, and that diameter is perpendicular to every line in the family.

6:55Both halves of that are facts about circles, and not about parallel lines. Take an ellipse instead. A family of parallels still has exactly two touching members, and they are still opposite each other through the centre. But the line joining them is not square to the family any more. Circles are the shape where those two things happen together. The single shared point of a tangent has a name. It is the point of contact.

7:27And now touching means something you can check rather than something you judge by eye. Touching is sharing exactly one point. Exactly one. Not nearly one. At radius five, set the distance to four point nine nine nine nine nine nine and the line still shares two points. There is still a chord there, tiny but real. Come from outside. Five point zero zero zero zero zero one, and it shares none.

7:59A millionth of a unit either way and there is no tangent in sight. Only the exact equality gives one. That is worth being fussy about, because everything proved about tangents from here on leans on the shared point being exactly one. You have seen this one without being told it was mathematics. A wheel resting on a road. The road is a straight line, the rim is a circle, and they share one point. The road is a tangent to the wheel, and the place they touch is the point of contact.

8:37Draw the spoke from the hub down to that point and it looks like it stands square to the road. It does, for a wheel on a level road, and that is easy to see, because the lowest point of a circle sits directly under its centre. But look at what has just happened. I picked a horizontal road because it made the answer obvious. The real claim is about any tangent at all, at any tilt: the radius drawn to the point of contact stands square to the tangent there.

9:11That is not obvious, and nothing in this video has proved it. That is the next one's job. So let us be honest about what these two experiments did. They showed that a tangent is a secant whose chord has run out, and they showed it twice, once by turning and once by sliding. They gave us a tangent at a chosen point of the circle, so we know at least one exists there.

9:39They made it look as though there is only one tangent at that point, because every other position of the wire crossed the circle twice. But looking is not proving. Every position of one wire on one table is a demonstration, and a demonstration cannot rule out the case you did not happen to try. Uniqueness is still owed, and it gets paid properly later, from the right angle we just refused to assume.

10:11What you can take away now is the sentence this whole topic turns on. A tangent is not a line that touches without crossing. It is a secant whose two crossings have become one point, and the chord between them has run out.