Properties of a rectangle, and why the square is the special case

0:00A rectangle is a four-sided shape whose corners are all square. That is the whole of it. Not the matching sides, not the parallel sides, not anything about the diagonals. Just the four corners. Everything else a rectangle does was demoted, and the demotion is the interesting part. Those things did not stop being true. They stopped being part of the description. A consequence is something you no longer have to ask for, because asking would be redundant.

0:35So: one condition in. What comes out? Four things, in the usual telling. We are going to count them again at the end, and get a different answer. First consequence: opposite sides come out the same length. Join one corner to the corner diagonally opposite, and the shape falls into two triangles. They share that line, so one side of each is the same length without anyone measuring it. The diagonal splits a square corner into two parts, and the angle at the far end has to make up the rest of a straight line's worth.

1:15Those two work out equal, and two angles with the shared side between them force the triangles to be identical. Identical triangles have matching sides, and those sides are the opposite sides of the shape. You asked for four corners. You were handed the sides. Second consequence: opposite sides are parallel. Take one side and follow it across the two sides it meets. It makes a square corner with each of them, and two square corners come to a hundred and eighty.

1:51That total is the test for the two lines it crossed being parallel. So they are. But be careful what that test detects. Lean the shape over until nothing is square any more, and the same two angles still total a hundred and eighty. The test knows about parallel. It knows nothing whatever about right angles, and that will matter shortly. Third and fourth: the two diagonals are the same length, and each one cuts the other exactly in half.

2:25Those are the two facts a carpenter uses in reverse to build a rectangle out of two sticks and a pin. Here they come out the other way round: build a shape with four square corners and the diagonals arrive already behaving. So the list is four properties long: matching sides, parallel sides, equal diagonals, halving diagonals. All four bought with one condition. Every four-cornered shape in a sweep of seven thousand six hundred and twenty-two has all four, and not one fails any of them.

3:04Take the condition away and each of them breaks somewhere. So all four are genuinely being paid for. That is not the same as all four being different. Here is what happens when you check them shape by shape instead of reading the list. Go through the whole sweep and ask, of every shape: do matching opposite sides and halving diagonals ever give different answers? Never. Not once in seven thousand six hundred and twenty-two.

3:37Ask the same of matching sides and parallel sides. Also never. Those three are not three properties. They are one property, said three ways, and the thing they all say is that the shape is a parallelogram. Equal diagonals is the odd one out. It disagrees with the other three on plenty of shapes, because a leaning shape can have matching sides and unequal diagonals. So the list of four is really a list of two, and only counting showed it.

4:13Now four shapes, every corner marked square, with their sides given. Five and two. Three point six and six. Five and one. And four, four, four, four. Which of them are not rectangles? The honest answer is that all four are rectangles, and most people hesitate over the last one. It has four equal sides, so it is a square, and being a square feels like being something else. But the requirement was four square corners, and it has four square corners.

4:47Nothing in the requirement said the sides had to differ. So define the square properly: a four-sided shape with four square corners and four equal sides. That is the rectangle's condition, plus exactly one more. Which means a square walks in already carrying every consequence a rectangle has, without a single new argument. Matching opposite sides, parallel opposite sides, equal diagonals, halving diagonals — all of them, free. You do not re-derive anything. You inherit it.

5:26And this is the general shape of the move: adding a condition never costs you a consequence. It only ever adds. People still resist this, and the resistance is worth naming. It feels as though a shape has to be one thing. Square or rectangle, pick one. But things belong to more than one family all the time, and nobody finds it strange anywhere else. The picture that settles it is two regions, one inside the other.

6:00Draw a region for the rectangles and a region for the squares, and the squares sit entirely inside. In the sweep, five shapes are squares and every one of them is also a rectangle. Not one escapes. And twenty are rectangles that are not squares. That is why the inner region is smaller, and why it is inside rather than beside. Now put the carpenter back to work. Two sticks the same length, pinned at the point that halves both.

6:34Whatever angle you open them to, the thread round the four ends is a rectangle. That is already settled. So the question is narrower than it looks: is there an angle that also makes the four sides equal? Try every whole crossing angle from one to a hundred and seventy-nine. Exactly one works. Ninety degrees, and nothing else. Cross the sticks square, and the rectangle you were always going to get is a square.

7:05With eight centimetre sticks, that square has sides of about five point six six. Once the sides are equal, two more things follow that a rectangle never had. Look at the point where the diagonals meet, and at the two triangles standing on one half of a diagonal. They share that half. Their other sides are two sides of the square, equal. And the other two are halves of the same diagonal, also equal.

7:36Three pairs of matching sides is enough on its own: the triangles are identical. So the two angles they make at the crossing are equal, and those two angles sit along a straight line. Two equal angles making a hundred and eighty between them are ninety each. The diagonals of a square cross at a right angle, and across the whole sweep that value never varies. And the second one. Take a diagonal and look at the corner it starts from.

8:10It cuts that square corner into two parts, and in a square those two parts are forty-five and forty-five. Every corner, every square, every size. Now do the same on a rectangle whose sides are not equal. The diagonal still cuts the corner in two, but never into halves — not once in the whole sweep. And the crossing angle, which is fixed at ninety for every square, takes a different value for almost every rectangle you try.

8:43Those two really are new. They are what the extra condition bought. So the picture is this. One condition gives you a rectangle and, with it, two genuine consequences wearing four names. One more condition gives you a square, which keeps all of them and earns two of its own. That is what a special case is: not a rival, not an exception, but the same thing with something extra required.

9:14It is also worth noticing what has not been asked. Nothing here says what happens if you demand equal sides and drop the right angles instead. That is a different shape with a different name, and it is not covered by anything argued here. Which is the honest place to stop: knowing exactly which questions you have answered.