The parallelogram: everything that follows from "opposite sides parallel"

0:00Take a rectangle and push it over. Hold the top edge and slide it sideways, keeping the two upright sides parallel. The corners are not square any more. Nothing else about the picture looks obviously broken. The question is which of the rectangle's habits survived that push, and which you have just thrown away. It is easy to assume they all go together, that being square is what holds a rectangle's other properties up.

0:31Most of them survive. One of them does not, and the one that does not is the whole difference. So it is worth being precise about what is left after the push. What is left is a single condition, and it is about direction only. Opposite sides parallel. Both pairs. That is the entire definition. Notice what it does not say. It says nothing about how long any side is.

1:01It says nothing about how big any angle is, or whether the figure is wide or thin, upright or leaning. That is a startlingly small thing to assume. Because everything for the rest of this comes out of it. There is no second condition arriving later. A shape defined by one hypothesis, and a list of consequences. Before deducing anything, see where the shapes you already know sit inside this.

1:32A rectangle has both pairs of opposite sides parallel. So a rectangle meets the definition. It is not a different kind of thing. It is one of these whose angles happen to be square. Here is a family built by taking two side lengths off a grid and an angle between them, with nothing chosen to be neat. Three hundred and seventy-five of them. Of those, twenty-five have four square corners.

2:02And of those twenty-five, five have four equal sides as well. Squares inside rectangles inside these. Not three separate subjects — one, with conditions added. Take one to work with, and build it the way you would on paper. Draw a side four centimetres long. From the same corner draw a second side five centimetres long, opened out to thirty degrees. Three corners so far, and no fourth one. Nothing has been assumed about where it goes.

2:37Through the end of the five, draw a line in the direction of the four. Through the end of the four, draw a line in the direction of the five. Wherever those two lines cross is the fourth corner. It is found, not chosen. That matters: open the two sides out flat and the two lines you draw are the same line, and there is no fourth corner to find.

3:03So the construction can fail, which is how you know it is doing work. Now the first deduction, and it needs one fact about parallel lines. When a line crosses two parallel lines, the two angles it makes on the same side of itself total a straight angle. Take one side of the figure as a line crossing the other two. It crosses two parallels, so the two corners at its ends total one hundred and eighty.

3:36In the figure just built, one corner is thirty, so the one next to it is a hundred and fifty. Every side does the same job, so every pair of neighbouring corners totals a straight angle. Across all three hundred and seventy-five, all four pairs, every time. That is a fact about one drawing. Making it a fact about all of them takes a letter. Call one corner x. Say nothing else about it — do not decide whether it is thirty or a hundred and twenty.

4:12The corner next to it is one hundred and eighty minus x. And the corner next to that one is a straight angle minus that, which is x again. So the two corners facing each other across the figure are equal. Not nearly equal. The same letter. Checked against the measured figure at every angle from twenty to a hundred and sixty: five hundred and sixty-four angles, and the letter argument gets every one of them right.

4:43It never had to know which parallelogram it was talking about, which is exactly what makes it a proof. It is fair to ask whether any of that really came from the hypothesis, or whether four-sided figures just behave that way. So take the hypothesis away and change nothing else. Run the same grid, with the same three corners each time, and move only the fourth one — off the crossing, by a little.

5:14Three hundred and seventy-five figures. Still four straight sides, still four corners. Neighbouring corners totalling a straight angle: none of them. Facing corners equal: none of them. Not fewer. Not most. Zero, from a family that differs by one corner having been nudged. Run the letter argument against those broken figures and it is right about only half of what it says — the half it copied from the hypothesis.

5:48Now the sides, and this needs a different move. Draw a diagonal. It cuts the figure into two triangles, and both of them have that diagonal as a side. So they already share one side exactly — the same segment, eight point seven of it in the figure we built. One pair of their angles is equal because those are the facing corners we just proved equal. A second pair is equal because the diagonal is itself a line crossing two parallels.

6:23Two angles and a side settle a triangle completely, so the two triangles are the same triangle drawn twice. Which means the sides opposite each other in the figure are equal — on all three hundred and seventy-five, and on none of the broken ones. One thing about that step is easy to get wrong on paper. Saying two triangles match is not only about the triangles. It is about which corner of one goes with which corner of the other.

6:57Write those corners down in the wrong order and the claim becomes false, even though the two triangles have not moved. Checked: swap two corners in the second triangle and the match fails, both ways of swapping. And matching angles alone is not enough either. The same shape at twice the size has all three angles equal and is not the same triangle. Same shape is not same size. The order and the lengths are both part of what is being claimed.

7:32Draw the other diagonal as well, so the two of them cross. Two more triangles meet at that crossing, and the same kind of argument works on them. Two pairs of equal angles from the two sets of parallels, and between them one equal side, from the facing sides just proved. So those triangles match too, and the crossing point sits exactly halfway along both diagonals. Each one cuts the other in half. On all three hundred and seventy-five, and again on a second sweep of two hundred and forty.

8:10On the broken figures, none. That is four properties out of one hypothesis, and none of them was assumed. Now the interesting part, which is the property that does not come out. Measure the two diagonals of the figure we built. One is eight point seven. The other is two point five. They are not close. Nothing in the argument ever suggested they would be. Across the sweep, the ones whose diagonals match are exactly the twenty-five with square corners, with no shape the two answers disagree about.

8:49Which makes it tempting to say equal diagonals and square corners are the same condition. Among these, they are. Step outside and they come apart at once: here are twenty-five figures with one pair of parallel sides and two equal slanted ones. Every one has diagonals of equal length. Not one has a square corner. The two answers disagree about all twenty-five. So here is the shape of the whole thing.

9:23One hypothesis about direction, assuming nothing about length and nothing about angle, and four properties fall out of it. Facing angles equal. Neighbouring angles totalling a straight angle. Facing sides equal. Each diagonal halved by the other. The fifth thing — diagonals of the same length — does not fall out, and that is not a gap in the argument. It is the gap between this and a rectangle. Add it back and you have added square corners.

9:57Which is why the rectangle sits inside: it is one of these with one more condition, and the square is one of those with one more again. A definition is not a description. It is the smallest thing you have to assume before the rest stops being optional.