 In this video we're going to state and prove the between cross lemma of order geometry and has to do with angles and particularly interiors of angles. So imagine we have an angle, the angle ABC, which just so you're aware, the angle ABC means that we have three non-colonial points. We have a, we have b, we have c. They're not collinear, so there's no line that holds all three of these points. And imagine we have a point P that is on the line AC. So we'll think of connecting the dots here. We have A and C right here, this line, line AC. And so imagine that P is some point on this line in an order geometry. Then we claim that P belongs to the interior of the angle ABC if and only if P is between A and C with respect to the line there. And so the diagram seems to suggest that. But one thing we have to be very careful about in order geometry is that we don't make assumptions based upon the diagram because it's really hard to draw diagrams that don't satisfy the betweenness relationships we're trying to prove. Because the diagrams we draw are typically or inherently Euclidean for which Euclidean geometry is an order geometry. So if you draw Euclidean picture, then it'll satisfy these conditions, but we're trying to prove it for any geometry. Think of like the real project of plane. So it's a lot harder to draw without like without like wraparound and things like that, right? If you start throwing wraparound into the game, it can get a little bit more confusing notions of betweenness fall apart. So don't rely too much on the picture. We're only going to draw pictures to try to motivate what's going on here. But the between cross limit tells us that P is between A and C only if and only if it's inside of the angle, which remember what it means to be inside the angle is that it's in the intersection of two open half planes. So let's go the first direction. Let's assume that P is inside the interior of the angle ABC. So to be to be inside the interior, it means that P is in the open half plane determined by the line BC that contains a, but it's also in the open half plane. Determined by the line BA that contains C like so. So P is inside that. This is the definition of this set right here. So in particular, we know that P and A are on the same side of the line BC. We also know that C and P are on the same side of the line BA. So that's that's what it means to be inside those open half planes. So I want to kind of unravel those for a little bit more here. So let's go with the first one. So we know that P and A are on the same side of the line BC. So we might get something like this. Here's our C. Here's our B. And so we have our point A maybe over here and P is over here. So we know again by assumption by assumption here, this will be the assumption throughout the whole thing, right? P is on the line AC. So the line segment PA doesn't intersect. It doesn't intersect the line BC. That's what it means to be on the same line. So when we look at this line, there's only a couple options we can have about the betweenness. It has to be that A is between P and C or P is between A and C. Now the second option is the one I've drawn on the screen. But if we just look at A and P being on the same side, it could be that P is over here. And so we have something like this. In particular by trichotomy, there's three options. The third option would be that C is between A and P. But if that were the case, that would imply that P and A are on different sides of the lines that can't happen. So then by trichotomy, we have these two options. A is between P and C or P is between A and C. But what if we switch directions a little bit? What if we now consider the line BC of BA? And we know that we have our point C, we have P, something like this. So we know that P and C are on the same side of the line BA. So we can have something like this. It could be that P is between A and C like we've drawn here. It could be that actually C is between P and A. Both of those would lead to C being on the same side of P. The other possibility by trichotomy, because one of those things has to happen, you could have... So we've already considered that C could be between A and P. We have P could be between C and A. The other possibility, of course, is that A is between C and P. But again, that would imply, like if P was over here somewhere, that would imply that C and P are on opposite sides of the line. That violates our assumption. So we can't have those either by trichotomy. So I want you to compare these statements that we have right now. So we either have PAC or APC, that's the first one. The second one tells us we either have PCA or CPA, which by the symmetric axiom between us, we can flip these things around. So I want you to be aware that these two are actually the same relationship. It's just one backward. By trichotomy, if you have this, that's going to be in violation of this one, because that's just the same one right here. You also are in violation of this one. There's only one relationship between PC and A. So you can't have that within forces this to happen. Similarly, you can't have this one. It's going to have to be the case that you have the P is between A and C. So in particular, that tells us that P is between AC, that is, it's on the line segment AC. That's exactly what we're trying to prove right here. So if P is in the interior of the angle, that means P is between A and C. So that's the first direction we're trying to prove for the between cross lima. So going the other direction, now let's assume that P is in the interval A to AC here. So I'm really thinking of this as the open interval, right? I know it's not A, I know it's not C, because really we're assuming that P is between A and C. So equality is not allowed in that situation. We didn't need to prove that P is in the open angle associated to the points ABC. So consider our three points. We of course have the points BA as we were drawing earlier. Let's say B was right here and the line A is right here. And then of course we had BC, something like this. So let's say C is the point right here. And then of course we're considering this line L that goes between A and C like so. But then again, that's not the line we're talking about. We have the line AC and then we have BC right here. So we know their point of intersection is this unique point C right there. So we're now assuming that P is between A and C, right? So clearly the diagram seems to suggest that P is in the interior, but we're trying to prove that the diagram is reliable in this situation. So because the intersection between the line AC and BC is the point C, if you look at the subset of the line AC and particularly look at the interval AP, if AP were to intersect the line BC, it would have to happen at C, which would then suggest that C is between A and P, which is not the case. That's not the assumption we have right here. So that tells us that AP does not intersect the line BC. And so in particular AP are on the same side of the line BC. So that's going to give us that P is inside the open half plane associated to the line BC and the point A. Okay, that's what we want here. Now by similar reasoning, the lines BA and CA will intersect at the unique point A like so. And so if you consider the line segment PC, if PC, which is a subset of the line AC, if you were to intersect the line BA, it would have to be at the point A, because intersections of lines are unique. That would imply that A is between PC, which is not the case, because trichotomy will only have one between this relationship and we know that P is between A and C. Therefore we have that CP are on the same side of the line BA, and that would then show that P is in the open half plane associated to the line AB, that's on the same side of C. So therefore P is in this half, this open half plane, it's inside this open half plane, the open interval, excuse me, the open angle is the interior of the angle, the intersection of these two, so P is in there, and that then proves the between cross lima.