The right triangle hidden in the figure, and the single number that sets the shape

0:00Last time we ended with three lengths and nothing connecting them. Half the long axis, called a. Half the short axis, called b. And the offset from the middle out to a fixed point, called c. Three names, no stated relation - deliberately, because each was found by searching the set of places, never computed from the others. By the end of this, one relation will connect all three. a squared equals b squared plus c squared.

0:32There is a right triangle in the picture, so it is tempting to file that away as an old theorem and move on. The triangle is used inside the argument. The triangle is not the argument. The definition says one thing. Take any place, measure to both fixed points, add, and compare with the constant. That question has the same answer everywhere on the curve, so we get to choose where to stand.

1:04Here is the good choice. Stand where the curve crosses the line running through both fixed points. Why there? Because that place sits on the same straight line as both of them. The two distances are not slants. They are pieces of one line, and pieces of one line add by looking. On a set with its fixed points four either side of the middle and a constant of ten, that line was searched step by step. It returned two places, minus five and plus five.

1:37So half the long axis is five. Found, not computed. Now the two distances from that place. Walk from the far fixed point. You cross the middle, which is c, and carry on to the end, which is a. That distance is a plus c. Walk from the near one instead. You are past the middle by c, with a still to go, so what is left is a minus c.

2:05Here a is five and c is four. The squared distances come out at eighty-one and one, so the distances are nine and one exactly. And the middle is doing real work. Search outward from a fixed point instead and you also get two places, at minus nine and plus one - but not one either side of where you started. Only the middle has the curve symmetric about it.

2:35Add the two distances. a plus c, plus a minus c. The offset appears once with a plus and once with a minus. It cancels. What is left is two a. Nine and one add to ten, and two a is ten. The definition never said what its constant was. It left that constant anonymous, and we then named half the long axis a on our own authority. This computation is the bridge. The definition's anonymous constant is the long axis length. Not similar to it. It. Every later use of two a rests on this cancellation.

3:17And it is not luck. Nine and one are not equal, so the sum coming out at twice the half-length is the offset dropping out, not two equal halves adding up. Along that whole line, other places where the two distances add to the constant: zero. That gives us a name for the constant and nothing else. For a relation we need the same constant computed again, somewhere the answer looks completely different.

3:49So go to the top of the short axis, directly above the middle. Finding that place is harder, and how it was done matters. Searching upward for a height that lands on the curve only works when that height is a fraction. Often it is not. So the search runs over squared heights instead. A place at height h above the middle is the same squared distance from both fixed points, and that is a question about h squared alone.

4:21Asked that way, the height is found even when the height itself is no fraction at all. For our set the search returns one squared height, nine. The short half is three. Now the two distances from up there. These are not pieces of one line. They are slants, one down and left, one down and right. But they are equal, and the reason is symmetry. Reflect everything in the vertical line through the middle. The two fixed points sit at equal offsets either side of it, so they swap, while our place stays put.

4:59A reflection does not change lengths, and two lengths that swap and are still the same two lengths must be equal. Measured, both squared distances come out at twenty-five. Both slants are five. That is no accident of these numbers. Across forty-eight sets, built at two different middles with the fixed points laid along four directions, sets where the slants came out unequal: zero. So both slants are the same length. Which length?

5:33Look at the triangle made by the middle, the top of the short axis, and one fixed point. One side has length b, one has length c, and the two are square to one another, because the short axis is square to the long one. So there is a right angle at the middle, legs b and c, and the slant is the hypotenuse. Now the old theorem enters, and this is the only place in the argument where it is used. The hypotenuse squared is b squared plus c squared.

6:08In numbers, nine plus sixteen is twenty-five, and the slant is five, exactly what the measurement gave. Across the family, sets where a slant was not the two squared legs added: zero. So the sum at this second place is twice the square root of b squared plus c squared. Before we finish, the shortcut, because almost everybody takes it: there is a right triangle with legs b and c, so its hypotenuse is a, so a squared equals b squared plus c squared.

6:44That reasoning is wrong, and here is the test. Keep the two fixed points where they are, and stand at any height above the middle, not just the top of the short axis. Every one of those heights gives a right triangle. Legs the height and the offset, right angle at the middle. Twenty heights were offered. The number making a right triangle is twenty. All of them. The number the definition accepts as a place on the curve is one. The height three, and nothing else.

7:18The triangle stands at every height, and says nothing about which one is right. What picks out the right one is the definition. Take that away and the relation vanishes; the triangle does not. Now put the two computations side by side. At the vertex the distances added to two a. At the top of the short axis they added to twice the square root of b squared plus c squared.

7:48Both are the same constant, because the definition holds one constant fixed for the whole curve. So they equal each other. Divide by two, square both sides, and there it is. a squared equals b squared plus c squared. What did the work? Two places on one curve, one definition applied at both, and an insistence that the two answers agree. The right triangle was one step inside the second computation. The argument is the agreement.

8:23And here the relation is a measurement, not an input. Across all forty-eight sets, both halves found by search and never from each other, sets where the long half squared was not the other two added: zero. Six distinct triples of squared lengths came out of that family, over three different sizes. Three things fall out. First, a beats b, because a squared is b squared plus something positive. Across the family, forty-eight out of forty-eight.

8:59Second, a beats c, by the same reasoning with the other term. Again forty-eight out of forty-eight. Third, and this closes a gap we left open: the axis through the two fixed points really is the longer one. We had been calling it the major axis before we had any right to. Forty-eight out of forty-eight. It also rules something out. A second relation looks almost identical, with the offset squared on the left. That one belongs to a different curve, and mixing the two up is the commonest error here.

9:38Across our forty-eight sets, ones where the offset squared was the other two added: zero. If you want a hook, make it structural: a is the biggest of the three, so a sits alone on the left. One more consequence, and it is the useful one. Before the relation the curve had three independent lengths. After it, b is not free: give me a and c and b is settled. Two independent numbers, not three - one sets the size, the other the shape.

10:12Split them apart by dividing one length by another. The offset out to a fixed point, over the distance out to a vertex: that ratio is the eccentricity. For our set, four fifths. One length over another, so it carries no units. It is a bare number. How big can it be? Nobody told us, so count. Sets whose ratio sits strictly between nothing and one: forty-eight out of forty-eight, with four distinct ratios from five thirteenths up to twelve thirteenths.

10:49Now watch what it controls. Hold the long half at ten and slide the fixed points outward, nine steps. The squared short half falls from ninety-nine down to nineteen. The ratio climbs from one tenth to nine tenths. Distinct long halves across the whole sweep: one. The size never moved. Only the shape did. Slide the fixed points all the way together and the offset becomes nothing. The argument then has nowhere to stand: there is no line through two points that are the same point.

11:27The set is still there: twelve places of a small grid, all at the same distance from that single point. It is the argument that stops, not the curve. Two sets, side by side. The first has a long half of five and a squared short half of nine. The second, ten and thirty-six. Every length in the second is twice the matching one in the first. Their ratios? Four fifths, and four fifths. Identical.

12:00So a bigger ratio does not mean a bigger curve. A third has long half thirteen and squared short half a hundred and forty-four - larger than both in every length, with a ratio of five thirteenths, smaller than either. Bigger and rounder: size and shape are separate questions. Finally, reading the three lengths off four questions. The first gives twenty-five and nine, so the squared offset is sixteen and the squared ratio sixteen twenty-fifths.

12:34The second must be divided through first, giving nine and four with a squared offset of five. Its long axis stands upright, and that offset is not whole. The third arrives as vertices and fixed points rather than an equation: thirteen, twelve and five. The fourth has squared halves of a hundred and of seventy-five, so that one is not whole either, and its ratio is exactly one half. Of the four, ones whose short half is a rational length: three. Ones whose offset is: three. All four ratios sit strictly between nothing and one.

13:16Three of the four could have a set built and searched to check the reading against. Disagreements: zero. The fourth could not be reached, and that is reported rather than counted as a pass. Untidy numbers are normal. The relation does not care, and neither does the ratio.