The Corollary: a diameter stands on a right angle wherever you take the point

0:00Some results are earned and some are collected. There is a word for the second kind: a corollary. A corollary is a result that follows immediately from something already proved - no new argument, no new idea, just a substitution into a machine you have already built. Be careful here: corollary sounds like it means small, and it does not. It says nothing about how important the result is. It says only where the result came from.

0:33The one in this video is a corollary, and it is also one of the most used facts about circles there is. Knowing which results are cheap consequences of which is part of the mathematics, not bookkeeping around it. Here is the setting, and it is almost embarrassingly plain. A circle. A diameter across it, from A to B, through the centre. And then a third point somewhere else on the circle - call it D.

1:04Anywhere else. Not at A, not at B, otherwise wherever you like. Join D to A and join D to B, and you have a triangle sitting on the diameter. The question is the angle at D. And the answer, every single time, is a right angle. That is the result. Now watch how little it costs. The relation we are about to use does not talk about chords. It talks about arcs.

1:34So the first thing to do is say which arc. A and B cut the circle into two runs, and D is standing on one of them. Name the other one - the arc from A to B that does not pass through D. That is the requirement the relation makes: the point you are reading from has to be off the arc you named. Everything ahead is one substitution, so this is the only place there is a choice to get wrong.

2:02Hold on to it. We will come back to it, and it will not behave the way you expect. Now, what does that arc make at the centre? Put a radius along the one going out to A, and turn it along the arc until it lies along the one going out to B. Here is the whole thing. A and B are the two ends of a diameter, so those two radii point in exactly opposite directions.

2:31Turning from one to the other is turning through a straight angle. A hundred and eighty degrees. And notice it is a hundred and eighty rather than nothing, because the radius really does sweep - it does not just compare two directions at the end. That distinction is the whole reason the definition was set up the way it was. So now substitute. The arc makes a hundred and eighty degrees at the centre.

3:02The doubling relation says a point of the circle off that arc reads half. Half of a hundred and eighty is ninety. The angle at D is a right angle. That is it. That is the whole derivation, and it is one line of arithmetic on top of a theorem that was already standing. No new construction, nothing added to the figure, no case to check. Which is exactly what the word corollary was reserved for.

3:34Now slide D round the circle and watch what happens to it. The triangle changes shape completely - tall, squat, nearly collapsed - and the right angle does not budge. It cannot. Nothing in the argument mentioned where D was. The arc was fixed the moment A and B were chosen, so its angle at the centre was fixed, so half of it was fixed. This was checked at two hundred and twenty positions, and the reading was a right angle at every one of them.

4:07And here is the check that means something: every ordinary chord was tried too - four thousand four hundred cases - and not one gave a right angle. It is not that the diameter is a convenient choice. It is the only choice. Back to the arc we named, because something strange happens here. Suppose you name the wrong arc - the one D is actually standing on. You would expect a wrong answer. You do not get one.

4:39A diameter cuts the circle into two half circles, and both of them sweep a straight angle, so both of them halve to ninety. Across every position tested there was one reading in the whole family, either side of the chord. So the number survives the mistake. What does not survive is the argument, because the relation you quoted required the point to be off the arc, and it was not.

5:08Try that on an ordinary chord and the two sides never once agree. A step you can skip without being caught is the most dangerous kind there is. There is a second route, and it uses nothing but triangles. Join D to the centre. Now there are two triangles, and both are isosceles - two of their sides are radii, and radii are equal. So in the left one, the angle at A equals the angle it faces at D. Call it a.

5:42In the right one, the angle at B equals the angle it faces at D. Call it b. The angle at D is those two pieces together: a plus b. And the three angles of the whole triangle add to a straight angle - so a, plus a plus b, plus b, makes a hundred and eighty. Which forces a plus b to be ninety. The angle at D again, with no circles quoted at all.

6:12Now a connection that usually goes unmade. Here is a fact about right-angled triangles that looks unrelated: the point equidistant from all three corners sits at the middle of the longest side. That is the same statement as this one, read from the other end. If the centre is at the middle of that side, then that side is a diameter, and the angle facing it is the right angle we just derived.

6:43Solving for that point directly, rather than assuming it, it landed on the diameter's midpoint in all two hundred and twenty cases. For an ordinary chord it landed there in none of them. Two figures that are one fact. Run it backwards and it becomes a way of finding things. Take a segment ten units long, and ask where you can stand to see it at a right angle. The answer is the circle that has that segment as its diameter - so every such point is exactly five from the segment's middle.

7:21A hundred and ten such points were built and checked: all five from the middle, and all with their two distances squared adding to a hundred. That last part is the theorem about right-angled triangles falling out on its own. One member of that family has whole-number distances: six and eight. Thirty six plus sixty four is a hundred, which is ten squared. One warning, because this result is badly over-applied.

7:53Take a four-cornered shape with all four corners on a circle, and let one of its sides be a diameter. The word diameter appears, so the reflex is to write ninety and move on. Sometimes that is right: two of the angles do stand on the diameter, and those are right angles in every one of two thousand and ninety arrangements tested. But the pair you are usually asked about stand on a different chord altogether.

8:24Those two are merely equal to each other, and only when their corners are on the same side of that chord - which happened one thousand four hundred and thirty times, agreeing exactly when it should. Check which chord the angle actually stands on. Every time. One last use, and it is a pretty one. Put a rectangle inside a circle with all four corners on it. Each of its angles is a right angle, so each diagonal is seen at a right angle from the other two corners - which means each diagonal has to be a diameter.

9:04So the diagonals cross at the centre. Out of seven thousand three hundred and fifteen ways of picking four points, exactly fifty five made a rectangle - every one of them two diameters wearing a hat. Four of those were squares - the cases where the two diameters happen to be perpendicular. And all of it came from halving a straight angle. That is what a corollary is for.