 As a reminder in the previous video for lecture six, we introduced the four axioms of incidents, which we give them the names line determination, secant C, point exist, and non-colonarity. Line determination tells us that given any two points, they determine a unique line. That is, any two points, there's only one line. There's exactly one line on both of them. Secant C tells us that all lines have at least two points that could be bigger, but it has at least two. Point existent says there's at least three points in the geometry, there could be more. And non-colonarity says that the set of all points is non-colonial, that is, there's no one line that's incident to all the points. Any geometry that satisfies these four axioms is called an incidence geometry. So in this video, I wanna start developing the theory of incidence geometry. I'm gonna prove some theorems, and these are theorems that'll be true for every incidence geometry that there is. We mentioned how 3.4.5 point geometry is, these are all examples of incidence geometry. Euclidean geometry is incidence geometry. Stefano, young geometry, these are all incidence geometries. Incidence geometry is a very broad, very diverse theory of geometries. A lot of geometries can be incidence geometries. It's somewhat like the most fundamental geometry, at least from our perspective. Of course, there are non-incidence geometries that exist, but we really aren't gonna care about them. For us, if you're not an incidence geometry, we kind of only think of you as a counter example. Don't care about you. And so these theorems we're gonna develop in this video and then the next video are gonna be exclusive, well, let me say that again. These are gonna be properties true for every incidence geometry. So this will be true for time and more I'm going forward. Every geometry we talk about in the future will basically have these properties, as long as it's an incidence geometry, but like I said, rarely, if ever, when we talk about something that's not an incidence geometry, we mostly will do those for the sake of counter example. So the first theorem of incidence geometry, if two distinct lines intersect, then the intersection is exactly one point. We have seen this theorem before in a couple of different incarnations. If two lines intersect, the intersection is unique. That's what we want here. So in the variations, we didn't have the if there. Like for example, with phantogymtree, we didn't have the if because it had an action that said all lines intersect. So there was no conditional there. This statement could be vacuously true because the condition might not be satisfied. So whatever about the conclusion. But if two lines intersect, then the intersection is exactly one. And I claim that this statement is essentially just the contrapositive of line determination axiom. By assumption, two lines have at least one point of intersection. So we say the intersect. So their intersection is at least one point. So we have our two lines, L and M. And so they have at least one point of intersection. If there were a second one, so if we assume, if we, proof by contradiction here, let's assume that intersection is not exactly one. There's some other point where the intersect call that point Q. Then we would have two lines determined by P and Q. Cause remember line determinations has given any pair. Our pair is gonna be P and Q. There is a unique line incident to both of them. We have two. So this contradicts axiom one, I one, it contradicts line determination. And so therefore we get a contradiction. So our assumption was wrong. The assumption was that they intersected at a second point. So the intersection, if there is one, must always be unique. This only used line determination. No other axioms were necessary. So in fact, this theorem, this first theorem Vincent's geometry is really just a theorem of line determination. If a geometry has line determination, then intersections are unique when they happen. Now in phano geometry, I mentioned how we scratched the hypothesis here because in line, with phano geometry, we have line determination that was axiom four, but we also said all lines intersects. That's why we were able to get that. We could strengthen that theorem. But in general, it was just this theorem right here. Axiom of the axiom of line determination gives us uniqueness of intersection. This is the main reason why we like line determination so much. It's this fundamental relationship between points and lines. Lines should only intersect at one point. If they intersect, line determination gives us that. All right, let's look at another theorem of incidence geometry. These are mostly gonna be very short. They're short proofs, I should say, following from one or two axioms, right? And we're gonna be stacking top each other. We're gonna be building this theory as we go along. So in line, in an incidence geometry, if L is any line, then there exists exactly one point, such that P is not on the line. Now, this is something we actually proved for young geometry, because we needed it. And we've observed this before. We've proved it before as well. This is a theorem of incidence geometry. If you have a line, there is a point not on the line. Now, I want you to be very careful about what this is saying here. I don't want you to confuse this with what axiom four said, the non-colinearity axiom. Non-colinearity says not all points lie on the same line. So if L is a line, then there exists at least one point not on it. So that's not saying the same thing as non-colinearity, but it's just one step away from what non-colinearity is telling us. Because after all, non-colinearity doesn't specify the line. This one does, but specifying the line is not a problem whatsoever. So take our line L. If no points existed off of L, then all points would lie on a common line. That would contradict non-colinearity. So we get it very, very quick. This is an immediate consequence of the non-colinearity axiom. So I can specify the line. This line does not have all the points. Therefore, there's some point not on L. This is the form of non-colinearity we use more often. Given a line, there's a point not on it. And so then let's look at a consequence of that. In incidence geometry, if P and Q are both, if P and Q are two points and they are distinct points, we shouldn't just, whenever we list two points, we shouldn't just automatically assume they're distinct because that can lead to some logical problems. We should be specific if we want them to be distinct. P and Q are distinct points. If they're distinct points, then there exists a third point, R, such that the set P, Q, R is a non-colinear set. That is, P, Q, and R are not on the same line. Well, how do we do that? Well, the points P and Q are given by assumption. And by line determination, that's what axiom one I was. I won, I should say. Which we're really not gonna use this number system very much farther in the future, but I am gonna mention it right now. Line determination tells us that given our points P and Q, there is a unique line that is incident to both of them. Like so, I kinda miss my point here, but that's okay. You just draw it bigger, infill it in so no one knows that you missed it. No big deal, just draw the dot big. We also know that by the previous theorem that given this line, call it L, there exists a point not on L. Call that point R, okay? So R is not on, so what we know here is that P and Q are on L, right? By construction, and we know that R is not on L. Again, by the previous theorem here. Are these three points collinear or non-colinear? I claim they're non-colinear. If they were collinear, if there was one line that contained all of them, right? There's some line here, M, that contain all of them. This would be a violation of line determination because there's two lines that contain P and Q. There's also, you can think of it as a violation of the first theorem we proved in this video. The intersection between L and M is two points now. So yeah, that's a problem. No line can contain P, Q, and R, so therefore the claim is now established P, Q, and R are non-colinear. So what this theorem tells us is if you give me any two distinct points, I can find the third point that forms a non-colinear set. Essentially, I can form a triangle given two points. We haven't defined what a triangle is yet, but intuitively we can kind of understand where I'm going with there. All right, the next theorem here. The next theorem says that in incidence geometry, if P is any point, then there exist points Q and R such that P, Q and R are non-colinears. No line that contains all of them. I want you to see how this theorem is a little bit different, a little bit more general than the previous theorem. The previous theorem says give me two different points I can find the third point so that all three are non-colinear. This one says give me one point, give me one point and I can find two other points so that these three points are non-colinear. It's interesting, right? So let's start off with our point P, okay? By point existence, we haven't used this one yet. We've used line determination, we used non-colinearity. All of the statements we've been proving as their conditional statements, they're an if, then form, they could be vacuously true, right? Now we're finally gonna use point existence. By point existence, there's another point Q that's not P because I know that by point existence, there's at least three points. P is one of them, perhaps, but there's got to be at least two others. I need just at least one. Q is that point. Then we can apply the previous theorem because now that we have two distinct points, we can apply the previous theorem and get there's a third point R for which we've now formed a triangle. We have this non-colinear set, okay? Let's go even one better, right? With the next theorem, I can tell you that there exists a set of three non-colinear points. So what have we done here? You give me two points, I can form a set of three non-colinear points. Oh, I can do better than that. Give me one point and I can come up with a set of three non-colinear points. Now we're like, give me no points. You don't even have to give me anything. There exists a three non-colinear point set. So that's the thing is give me basically any collection of points I can always extend it into a non-colinear set. That's really what we're saying here. There exists a non-colinear set of three points. I should mention also at this moment that any pair of points is always collinear. If you give me only two points, that's always a collinear set because of line determination. But that's why three is of importance here. If you want to form a non-colinear set, you need at least three points and those things do exist. These previous theorems improvement are not vacuous. There do exist non-colinear sets. We're gonna combine the previous theorems. The previous theorems have used line determination, non-colinearity and point existence. We're gonna use point existence one more time. By point existence, there is a point P. Okay, well by the previous theorem, if you have a point, that point can be extended to a set of three non-colinear points. Voila, we now have our triangle P, Q and R. We have our non-colinear points like so. So this is sort of a major result for incidence geometries. In every incidence geometry, there exists a set of three non-colinear points. Now you'll recall that three point geometry is an incidence geometry. We talked about that in the previous video. The diagram I've drawn on the screen right now looks like three point geometry. Three point geometry has nothing more than the diagram you see right now. And essentially, that's what I'm saying here is that three point geometry is the smallest incidence geometry in existence. You have at least this structure. You have formed this triangle. Anything beyond that turns out is not gonna be a theorem of three point geometry because if it's not true for three point, excuse me, anything beyond this diagram cannot be a theorem of incidence geometry because if it's not true for three point geometry, which is an incidence geometry, it's not true for incidence geometries. So this set of three non-linear points is guaranteed for every incidence geometry. What I wanna do is start going the other way around now. What about lines? Can we have non-concurrent lines? So remember, concurrent lines mean that they intersect at a common spot. So what stops us from having something like this where all lines pass through the same point? That's not what non-colonarity says that given a line, there's a point not on the line. That is to say, not all points belong to the same line. But do we get the other direction? What about concurrency here? Does there exist a point which is on every line? That incidence geometry doesn't do that. So that shouldn't, that can't be a theorem, right? What do we have here? Can we force that this type of diagram never happens at the incidence geometry? And the answer is yes. So that's our next theorem here. Given a point, then there exists at least one line such that P is not on that line. So we get that P does not belong to L right here. So this is like the dual statement to what we did previously. That previously we stated that given a line, there's a point not on the line. Now we want, given a point, is there a line not on the point? I should say the point's not on the line. That's how we usually state this. So given the point P, like so, by a previous theorem on the previous slide, right, we showed a theorem 147, if you're following along with those notes here. In 147 we show that given any point, there exist two points, two other points called Q and R, which P, Q and R are non-colinear. So apply that theorem, we get this triangle that we've been constructing right here. And I should mention, I'm not gonna draw those segments there. I should mention that by line determination, P, excuse me, Q and R have a line determined between them. And since P is non-colinear to Q and R, this line cannot contain P. Because after all there's only one line that contains Q and R. And there's no line that contains all of the points P, Q and R. Therefore P is not on this line. And so we've now constructed a line for which P is not on. So you'll notice here that when we had a line that doesn't contain a point, we only get that directly from non-colinearity. There were the other one, a point that's not on a line, that took a lot more structure to get. We use this non-colinearity statement that we've developed here, but we didn't get this dual statement. Given a point there's a line that that point is not on. And so for the sake of this video, I'm gonna do one more theorem. There is another, there's more theorems of incidence geometry we're gonna do in the next video. But I don't want this video to get too long and we'll do that in the next lecture for which this is a natural stopping point for the sake of this video here. But for our final theorem of incidence geometry in this video, for each point there exists at least two lines containing it. So we always have something like this. That's where we're trying to show. There's always at least two lines containing that point. So take an arbitrary point called P in our geometry. By theorem 147 there exist two points column Q and R for which are so that the set P, Q, R is non-colinear, right? So there's no line that contains all of these ones. By line determination there is a line determined by P and R. There's a line determined by P and Q. And so this gives us two lines containing P. Are these the same line? No, because if these two lines were the same that would mean that P, Q and R are collinear which we know they're not. So this gives us at least two lines. There could be more, right? But with the incidence axis, this is all we know like a three point geometry, that's the best you can get. Every point is incident to exactly two lines. And so anything that's not true for three point geometry that cannot be a theorem of incidence geometry because three point geometry is an instance geometry. That's why that model is so important. Three point geometry is important because things that are false about three point geometry cannot be theorems of incidence geometry. In particular, we can't guarantee that points are incident to more than two lines because three point geometry doesn't have that property. And so this theorem right here which we're gonna conclude this video with uses that theorem 147 which we used several times in this video. You'll notice though that in this development what we've done is we've used line determination a couple of times. We use non-colinearity. We used point existence. And we use theorems that derive from some or all of those axioms. You'll notice that at this point we have never used secancy in any of our proofs. So all of the theorems we've done are actually theorems of incidence axioms one, three and four. We haven't used secancy yet. And the reason that this is our natural stopping point is that the next theorems we developed for instance geometry in the next video which is in the next lecture will then start using secancy. And so we've separated because of that indicator right now. So thanks for watching. If you learned something about incidence geometry please like these videos, subscribe to the channel if you wanna see more of these in the future. And as always if you have any questions, any of these videos please feel free to post them in the comments below and I'll be glad to answer them at my soonest convenience.