The rhombus, and what its diagonals do

0:00Here is a condition you can check with a piece of string. All four sides the same length. Nothing about corners, nothing about direction, nothing about which way anything points. It is a statement about how far, and only about how far. So it ought to leave the shape almost completely free. Ask most people to draw one and they draw a square, then perhaps a square tipped over onto its corner.

0:29Both of those have four equal sides. Neither of them is the general case, and the general case is stranger than it looks. Because a condition about length is about to force something about direction, and there is no obvious reason why it should. So let us build one without assuming anything. Start at a corner. Draw one side four units long, going right. Draw a second side from the same corner, also four units, opened to fifty degrees.

1:04Fifty is not special. It is simply the angle I picked, and any angle short of a straight line would have done. Now the fourth corner. Set a compass to the same four units and strike an arc from the end of the first side. Strike a second arc, same radius, from the end of the second side. Where the two arcs cross is the corner. It is found, not chosen. The two arcs decide where it goes, and the only instruction they were given was that length.

1:39Look carefully at what went into that. One length, used four times. One opening angle, chosen freely. At no point did I say the corners should match, or that any two sides should run the same way as each other. And here is the figure. The corners are fifty, a hundred and thirty, fifty, and a hundred and thirty degrees. So equal sides plainly do not make equal angles. Two of these corners are sharp and two are blunt.

2:11That kills the first guess. A shape with four equal sides is not obliged to be a square, or anything like one. Sweep the opening angle and the side length across seventy-five different figures and only five of them come out square. A four-sided figure with all four sides equal is called a rhombus. That is the entire definition. Four sides, one length. It is worth being careful about what that name does not carry.

2:45It does not say the corners are square. It does not say the figure is a tipped-over square. It does not say anything about facing sides running the same way as each other. Every one of those is an extra claim, and extra claims have to be earned. So the interesting question is not what a rhombus is. It is what a rhombus cannot avoid being. Draw one diagonal, from a sharp corner across to the other sharp corner.

3:17It cuts the figure into two triangles, and those two triangles have identical sides. Two sides of the rhombus on one, two sides of the rhombus on the other, and the diagonal shared between them. Three matching sides means the two triangles are the same triangle, drawn twice. And each of them is isosceles, because two of its three sides are sides of the rhombus. The base angles of an isosceles triangle are equal. So this pair matches, and that pair matches.

3:54Put the two facts together and all four of those angles are equal. On this figure each of them is twenty-five degrees. Now read those four marks differently. The diagonal is a line crossing two other lines. Two of the four angles sit on opposite sides of it, in the alternate position. Equal alternate angles is precisely the test for two lines running the same way as each other. So this side and the side facing it are parallel. Not because they look it, but because the two marks are equal.

4:33The other two marks are the other alternate pair, sitting at the same crossing. So the other pair of facing sides is parallel too, for exactly the same reason. Nothing here was assumed. The equal sides gave the equal angles, and the equal angles gave the direction. That is worth stopping on, because it is the surprise. A condition about length has forced a fact about direction. Both pairs of facing sides parallel is the definition of a parallelogram. So every rhombus is a parallelogram.

5:13Not as a resemblance. As a consequence, with no angle ever assumed anywhere in the argument. Which means the rhombus inherits, free of charge, everything that follows from being a parallelogram. Facing corners equal. Neighbouring corners adding to a straight angle. Each diagonal cut in half by the other. Test that across all seventy-five equal-sided figures and it holds on every single one. Here is what that inheritance buys you.

5:49Take the fifty degree figure. What are the other three corners? One route: use the isosceles triangle. Its three angles make a straight angle, and two of them are equal. So each base angle is a hundred and eighty take away fifty, halved. Sixty-five degrees. Two of those meet at the next corner, so that corner is a hundred and thirty. The other route is now available: neighbouring corners of a parallelogram add to a straight angle, so it is a hundred and thirty directly.

6:26Same answer. Two arguments, one from equal sides and one from parallel sides, and they agree at every opening angle. So where does this shape sit among the others? Draw one ring for figures with four equal sides. Draw a second ring for figures with four square corners. They overlap, and the overlap is not empty. A figure can have both. That overlap is the square, and it is a small part of either ring. Sweep them together and only five figures land in it.

7:02Both rings sit inside the parallelograms, and the rhombus ring sits there because of the argument you just watched. The square is not the starting point. It is the place where two separate conditions happen to meet. Now draw both diagonals and look at where they cross. They cut each other in half, because a parallelogram's diagonals do that, and this is a parallelogram. So take the two triangles on either side of one diagonal, meeting at that crossing.

7:38Two sides of the rhombus, equal. Two halves of the other diagonal, equal. The piece from the crossing outward, shared. Three matching sides again, so the two angles at the crossing are equal to each other. And they sit together on a straight line. Two equal angles filling a straight angle are ninety degrees each. The diagonals of a rhombus meet at a right angle. On this figure they measure seven point three and three point four units.

8:12So sort the list into what was inherited and what was bought. Inherited from being a parallelogram: facing corners equal, neighbouring corners making a straight angle, each diagonal halved by the other. Bought with the equal sides: the diagonals cross at a right angle. And bought with them too: each diagonal cuts the two corners it passes through into equal halves. That second one is just the four equal marks again, read once more. Twenty-five and twenty-five at one corner.

8:49On the other diagonal the four marks are sixty-five each, and they halve the blunt corners the same way. Only the last two needed the sides to be equal. Everything above them arrived with the parallel. One last thing, because the list invites a mistake. It is tempting to turn the right-angled crossing round and use it as a test. Diagonals crossing at ninety degrees, therefore a rhombus. That is false, and here is the figure that breaks it.

9:23Two pairs of equal sides, but equal to their neighbours rather than all four together. A kite. Its diagonals cross at a right angle. One of them even halves the corners it passes through. What it does not do is cut both diagonals in half, and its sides are not all equal, and its facing sides are not parallel. Copying some of the behaviour is not the same as having the condition. The condition is four equal sides, and everything else was earned from it.