 In this video, we're going to continue our development of the ordered structure, the betweenness structure of a geometry, assuming the four betweenness axioms of Hilbert. In particular in this video, we're going to prove the so-called linearity rules, which then give us way of extending betweenness relationships of points on a line to other betweenness relationships. So imagine we have a line, L, and we have four distinct points, A, B, C, D, which are distinct points on this line, and this is inside of an ordered geometry. So we have the incidence axioms, we have the betweenness axioms. And so the linearity rules tell us the following. If we know that B is between A and C, and we know that C is between A and D, these betweenness relationships can be extended into the fourth, the four layer between the statement A-B-C-D. Now this might seem obvious at first, but I want to remind ourselves, what does this statement A-B-C-D even mean? It actually means four betweenness relations. So this four area betweenness means that B is between A and C. It means that C is between A and D. It means that B is between A and D. It also means that C is between B and D, like so. So one of these four area between the statements actually means four different trinary between the statements. And so that's the importance of the linearity rules here that if I have two betweenness statements, this will imply the other two. Because so if we know B is between A and C and we know that C is between A and D, then we can infer that B is between A and D and C is between B and D. That's the first linearity rule. There are two other ones we want to state right here. We have that if we know that C is between B and D and we know that B is between A and D, then that means that B and C are both between A and D and particularly have A-B-C-D. It's the same statement, but you have a different set of hypotheses to get the final conclusion. And then likewise, if B is between A and C and C is between B and D, then we have that A-B-C-D holds as well. So again, same conclusion, different hypotheses. Now, I want you to be aware that any combination of two between the statements does not always imply this four area between this. For example, if we had a statement like, oh, B is between A and D and we know that C is between A and D, this does not imply A-B-C-D because the idea is we know that B and C are between A and D, but we don't know how B and C relate to each other. So if we look at the line, we have like A here and D here. Yeah, it could be that it goes A, B, C, D, but it could also go A, C, B with that information. So I want you to be aware that just because you know two between the statements does not automatically imply the other two statements, but at least with these linearity rules that are on the screen, if you know these two, these two, or those two, that all imply the two other ones you don't know. Okay? Now in this video, I'm going to prove the first linearity rule, and I'm going to leave it as an exercise to the viewer here to prove linearity rules two and three. They're going to be strikingly similar to the proof you're going to see of number one. That's on the screen. So by assumption, we know that B is between A and C, and we know that C is between A and D. We need to prove that B is between A and D, and we need to prove that C is between B and D. So that's our goal. So let's look at the first goal right here. B is between A and D. So the way we're going to approach this is we're going to consider a line. So this is our line L, like so. And we're going to consider a point E that's off of the line. Now by assumption, we know that B is between A and C. So I'm going to want to try to draw these on the screen here. But one has to be very careful. It's really hard to draw this picture without sort of circular reasoning. The way I draw them, it kind of forces it to be what I think is going to happen. So we have to make sure we don't rely too much on the diagram here. The diagram can be extremely deceptive. So E is some point that's not on L, and I want to consider the line between E and C. So if we were to connect those together, consider this line right here. Now necessarily, the line EC will intersect the line L at a unique point, and that point's going to be C. Order geometries in addition to the betweenness axioms also satisfy the incidence axioms. So we know that when two lines intersect, it intersects at a unique location. So now consider the following. So by assumption, B is between A and C. So if we look at the assumption, well, because B is between A and C, we know by trichotomy that C is not between A and B. And as such, because trichotomy only allows for one betweenness relationship between three points, and we have by assumption that B is between A and C. So in particular, why does this matter? If we look at the line segment A and B, so the line segment contains the points A and B, it also contains everything between A and B. This line segment, which is a subset of the line L, cannot intersect the line EC, because if it did intersect, because we have a subset of the line L, the intersection between A and B and the line EC would have to be the point C. But by trichotomy, C is not between A and B. This tells us that the line segment A and B doesn't intersect EC. Therefore, A and B are on the same side of the line of the line EC. So that's an important observation to know there. The other assumption we have is that AC is between A and D. I'm actually going to put back that little squiggle here. So we remember that we know that A and B are on the same side of the line here, the line EC. But we also know that C is between A and D. So I'm going to add another point here, D, like so. So if I consider the line segment, I'll use a different color for this one. If I consider the line segment AD, like so, by assumption, we know C is between the two. So this tells us that A and D are on opposite sides of the line EC. So these two points are on opposite sides of the line. Now we're going to get to the idea of plane separation. This is something we're going to use a lot when we talk about order geometries here. So by plane separation, since A and B are on the same side, and since A and D are on the opposite side, that tells us that B and D, Bravo Delta, are on opposite sides of the line. And since they're on opposite sides, that means the line segment BD intersects the line EC at some point between B and D. But again, BD is a subset of the line L, and L intersects EC only at one point, and that's C. So if the segment BD intersects the line EC, it must be at the point C. And so from that, we can then infer that C is between B and D. And so that gives us, of course, the very first of these two between this relationships that we're trying to do. Okay, I'm going to zoom out a little bit so you can see the picture, plus also the proof that we're arguing about. So we've now proven that C is between B and D. Why is A, excuse me, why is B Bravo between Alpha and Delta there? Well, basically you do the exact same argument here, but now we're going to consider the line that goes through E and B, like so. And you play the same game. By the assumptions we have, D and C will be on the same side of the line. We're going to have that A and B are on opposite sides. And then by plane separation, you're going to get that A and C are on opposite sides. And therefore, there's a point between them that's on the line EB. And by uniqueness intersection, that point's going to have to also be B. I went through that one really quickly, but it's basically the exact same argument. You just move your position a little bit. And so that then proves the first of the linearity rules. And like I said, I'm leaving it as an exercise to the viewer to prove the other two linearity rules. I promise you they're no more difficult than what we're doing right here. And thus it's a good exercise to understand these betweenness axioms to check these things out and see exactly why the other two linearity rules follow.