What the diagonals alone tell you about a quadrilateral

0:00Lay one straight segment across another so that they cross. That is all. Two segments, one crossing, nothing else placed. Now look at the four ends. Join them up in order. You did not choose those corners. The two segments chose them, the moment you laid the second one down. So a quadrilateral does not need its sides to be specified. Its two diagonals carry the same information, packaged differently.

0:31Which means you can describe a shape entirely by what its diagonals do, and never mention a side at all. The surprise is how few questions that takes. Here is the experiment. Two elastic bands of the same length. Cross them so they meet at a right angle, and slide them until each one passes through the other's middle. Pin the four ends down. Do not join them yet. Before you draw a single side, you already know what shape is coming.

1:04Say what it is, and say why, and only then join the ends. That order matters. If you join first and name afterwards, you are recognising a picture, not deducing anything. So: what must it be? The two segments are the same length. That is the first fact. Each passes through the other's middle. That is the second. They meet at a right angle. That is the third. Halving each other alone would give facing sides equal and parallel.

1:40Equal lengths on top of that force every corner to be square. And the right-angled crossing forces all four sides to come out the same length. So it is a square, and with bands of eight units each, its side is five point seven. Now you may join the ends. Now change one thing, carefully. Take one of the two bands and stretch it by two units at each end.

2:08At each end. Both ends, by the same amount, so the crossing stays exactly in the middle of it. That band is now twelve units long instead of eight. Ask the three questions again. Same length: no, not any more. Each halving the other: still yes. Meeting square: still yes. Exactly one of the three answers changed, so exactly one name changes. It is a rhombus now, with a side of seven point two.

2:39Sweep the stretch through every amount you like and it is a rhombus every single time, except the one time you do not stretch at all. Now do it wrong, deliberately, because the wrong version is where the detail lives. Stretch the same band by two units at one end only. It is ten units long. But the crossing is no longer in its middle — it sits four units from one end and six from the other.

3:10So a different answer changed. The other band still gets halved. This one does not. And now the shape is a kite: two pairs of equal sides, but equal to their neighbours instead of to the side facing them. Its two pairs measure five point seven and seven point two. The amount you stretched by was arbitrary. Which ends you stretched was not. So here are the three questions, and they are the whole video.

3:41Are the two segments the same length? Does each of them cut the other in half? Do they meet at a right angle? Three yes-or-no answers. Every named quadrilateral you know sits at one of those combinations. And none of the three is a restatement of another — you can build a figure with any one of them false and the other two true. Let us fill the table in, one row at a time.

4:11Start with halving on its own. Each segment cut in half by the other, and nothing else asked for. Different lengths, any crossing angle you please. Join the ends and the facing sides come out parallel, both pairs. That is a parallelogram, and it is the base of everything above it in the table. Four hundred and twenty different figures were built this way, across a grid of lengths and crossing angles.

4:40Every one of them a parallelogram. Not most. All. Halving is the condition doing the heavy lifting, and the other two questions are refinements of it. Keep the halving. Now also make the two segments the same length. Every corner becomes square, and you have a rectangle. Notice what did not have to be true: the crossing angle. Set it to thirty degrees, or forty, or a hundred and forty.

5:11You still get a rectangle every time. A different rectangle each time, but a rectangle. Eighty-four figures were built with equal halving segments, at every crossing angle in the sweep. All eighty-four rectangles. So equal plus halving is the rectangle's row, and the crossing angle simply chooses which rectangle. Go back and take the equal condition off again. Different lengths, still halving. This time make them meet at a right angle instead.

5:46Now the four sides come out the same length, and you have a rhombus. Thirty figures built that way, thirty rhombuses. Compare the last two rows carefully, because they are the same shape of statement. Halving plus equal lengths gives square corners. Halving plus a square crossing gives equal sides. The two conditions trade places, and so do the two conclusions. That symmetry is not a coincidence, and it is worth sitting with.

6:19Now ask for all three at once. Same length, each halved, meeting square. Square corners from the first two, equal sides from the last, and a square is what you get. Which is the first arrangement of bands, arrived at from the other direction. It is also why two diameters of one circle, set at right angles, give you a square with no further instruction. Two diameters are automatically the same length and automatically halved at the centre. Two of the three conditions come free.

6:56And it is why, with two sticks of equal length and a piece of thread, you can make an exact right angle with no protractor. Cross them, thread the four ends into a shape with four equal sides, and the crossing has to be square. One row is left, and it is the one that catches people. Meeting at a right angle, but with only one of the two segments cut in half.

7:24That is the kite, and it means the reverse claim is false: a square crossing does not make a rhombus. The rhombus needs both segments halved. The kite halves one. Thirty kites were built that way, and not one of them was anything else. So the table is not a list of examples. Put every figure against every row and exactly one row in five answers for it. Five hundred and seventy agreements out of two thousand eight hundred and fifty pairings. One row each, no shape claimed twice.

8:02Two warnings, and then a use for all this. Equal lengths on their own buy nothing. Take away the halving and not one figure in the sweep earns a name at all. A right-angled crossing on its own buys nothing either, for exactly the same reason. Put the halving back and a name arrives immediately in both cases, so those are absences that were looked for. And one shape has no row here at all. A trapezium is defined by one pair of parallel sides, and by nothing about its diagonals.

8:40So it is not that the trapezium was forgotten. It is that this table asks a different kind of question, and the trapezium does not answer it. Which leaves you something practical: hand someone two lengths and a crossing angle — seven, five, and a hundred and forty degrees — and they can build the shape without being told a single side.