Counting shared points sorts every line into one of three kinds

0:00A circle is not really a round shape. It is a rule. Fix a point, fix a distance, and take every point that far from it. What you get is a circle. The fixed point is the centre, the fixed distance is the radius, and a segment joining two points of the circle is a chord. That is everything we are bringing with us. Now drop a straight line onto the page beside it, and ask the only question worth asking.

0:32How many points do the line and the circle have in common? Not where they meet, not at what angle, not whether it looks like it is touching. How many points are on both. That is a count, and a count is a number you can check. Slide the line about, turn it, push it off the page and bring it back, and you will find you can get three answers and only three.

0:57None. One. Two. You will never get three, and by the end of this video you will know why not, rather than just having been told. Here are the three, side by side. In the first, the line passes clear of the circle. Nothing is on both. The count is zero. In the second, the line runs across the circle, entering here and leaving there. Two points are on both. The count is two.

1:28And in the third, the line just grazes the circle. One point is on both. The count is one. That is the whole classification, and notice what did the sorting. Not the picture. The number. You can look at these three and see three different pictures. What matters is that you can count zero, two and one. Each count gets a name. Share no point at all, and the line is called non-intersecting.

1:59Share two points, and the line is called a secant. The segment between those two points is a chord, and the secant is the line that carries it. Those are two different objects. The chord has two ends. The secant runs on forever in both directions. And share exactly one point, and the line is called a tangent. The order matters here. The count comes first and the name second, because the count is what you can check and the name is only what we agreed to call it.

2:35So why can there never be three? Suppose a line met a circle at three different points. Two points already fix a line completely, so the third one would have to sit on that same line and on the same circle. Here is why it cannot. Drop a perpendicular from the centre onto the line. It lands in exactly one place. Now walk along the line away from that landing point. Your distance from the centre grows the further you walk, and it grows the same way whichever direction you walk in.

3:11So the points at exactly the radius sit at equal steps on either side of that landing point. There are at most two of them, and there is nowhere for a third to be. That is not a picture, it is a count. Take the twelve points of a circle of radius five whose coordinates are whole numbers, and every one of the two hundred and twenty triples you can pick from them is bent, never straight.

3:39Do the same with a circle of radius twenty five and its twenty points. One thousand three hundred and sixty triples in all, and not one of them lies in a line. Now for the measurement that decides which of the three you get, before you draw anything. Call the radius r, and call the perpendicular distance from the centre to the line d. Take the point of the line under the centre, and step out along the line by an amount t. The centre, that foot and your new point make a right triangle.

4:16So your distance from the centre squared is d squared plus t squared. That point is on the circle exactly when that comes to r squared. Which means t squared is r squared less d squared. Read the three cases straight off that. If d is less than r, there are two steps that work, one each way, and the line is a secant. If d equals r, the only step that works is no step at all, and the line is a tangent.

4:48And if d is more than r, there is no such step. The line misses. Three cases, from one comparison, and no fourth is possible because a square cannot be negative. Put numbers on it. Radius five. Bring a line in at a distance of three from the centre. Twenty five less nine is sixteen, so t is four, and the two crossings sit four either side of the foot.

5:18The chord is eight. Push the line out to a distance of five. Twenty five less twenty five is nothing, so t is nothing, and the two crossings have slid together into one. The chord has shrunk to length zero. That is the tangent. Push it out to seven. Twenty five less forty nine is negative, and nothing squares to a negative number. There is no crossing at all. Notice that the chord did not vanish suddenly. It shrank, steadily, all the way down to nothing.

5:55Watch that happen properly. Take one line and slide it across the circle without turning it at all. The direction never changes. Only the distance from the centre does. The count starts at two, and the chord is at its longest when the line goes through the centre. As the line slides out, the chord shrinks. Every step out makes it strictly shorter than the step before, all the way along.

6:24At the moment the distance reaches the radius, the count drops to one. And past that, it drops to zero and stays there. Thirty three positions checked, and the count never once goes back up. It passes through one exactly once. That is worth holding on to, because it says the three cases are not three unrelated pictures. They are three stretches of one journey. Which brings us to the thing students get wrong most often.

6:57A line very close to the circle is not almost a tangent. There is no almost. Either the distance from the centre equals the radius, or it does not, and if it does not then the line is a secant or it misses. Take the radius five again. Set the distance to four point nine, and the count is two. Four point nine nine, still two. Four point nine nine nine nine nine nine, still two.

7:27Come from the other side. Five point one, and the count is zero. Five point zero zero zero zero zero one, still zero. A millionth of a unit out and there is no tangent anywhere in sight. Only the exact equality gives one. That exactness is not fussiness. Everything proved about tangents later leans on the shared point being exactly one, and if nearly tangent counted, none of it would hold.

8:01Two last things worth being careful about. First, a line does not have a kind. It has a kind relative to a circle you chose. Here is one line and three circles about the same centre. Against the small one, it misses. Against the middle one, it is a tangent. Against the big one, it is a secant. Same line, three answers. What changed was the radius, not the line.

8:30Second, none of this came from the drawing. It came from the circle being a constant distance set. To see that, take a shape that is not a circle. Take the points a fixed taxicab distance from a centre, going across and up rather than straight there: that is a square standing on its corner, and its edges are straight. Run exactly the same search along one of those edges and it finds six shared points on a single line.

9:01So a cap of two is not a fact about straight lines. It is a fact about circles. You have met the tangent in the world already. A rope hanging over the wheel of a well runs straight down on each side, and leaves the wheel at exactly one point on each side. Both strands are the same distance from the wheel's centre, and the centre sits exactly halfway between the two points of contact.

9:30That is two tangents, holding a load. So: three positions, three counts, three names, and one measurement that decides between them. Nothing, when the centre is further from the line than the radius. One, when they are exactly equal. Two, when the centre is nearer. The third case is the one everything later is built on, and now you can say precisely what it is: not a line that touches without crossing, but a line that shares exactly one point.

10:04Count first. Name second.