 Welcome back to our lecture series, Math 3130, Modern Geometries for Students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Misseldine. In this lecture 21, we're going to talk about the ideas of circles. Now, if you're following along with our textbook, Roads to Geometry by Wallace and Wes, this topic is actually left until section 4.5. In the chapter among Euclidean Geometry. But it turns out, well, there are some results in that section that are specifically about Euclidean Geometry. In our lecture series, I want to introduce these notions right now. Actually, these are notions about circles in neutral geometry, or even better yet, these are notions of circles in congruence geometry. We're just going to use the notions of incidents betweenness and congruence to define and prove these properties about circles. That's going to be the topic of lecture 21 here. First and foremost, what is a circle in a congruence geometry? Remember, a congruence geometry is a geometry that satisfies the four axioms of incidents, the four axioms of betweenness, and the six axioms of congruence. Suppose we have two distinct points in a congruence geometry, call one O, call the other one R. Then we define the circle, which we'll often denote by Greek letters like gamma in this situation. We define the circle centered at the point O and with the radius O, R, to be the set of all points P in the geometry, such that O, P is congruent to O, R. So the circle is a set of points which are a locus. That is, they satisfy some equation. We want all of the points that satisfy the equation, O, P is congruent to O, R, where O and R are considered fixed points and then P is allowed to vary. So that is what we define to be the circle. The circle itself, as we often draw them like this, the circle is the elements right here on this circumference. Now the things inside the circle we can talk about, these are referred to as the interior points of the circle, for which we'll typically denote that as gamma circle there. Common notation for us here to use a little circle here. I should also mention that if we need to specify the center and the radius when we define the circle, we might write something like gamma O comma O, R, something like that. But when the center and the radius are clear from context, we'll just call the circle gamma in that situation. Therefore, gamma circle here is gonna be the interior of the circle. We're looking for all points P such that the segment O, P is less than O, R. And this is the usual meaning of the circle here. We have an O, right? We have some other point R, in which case then we have this radius, it's the segment O, R. The circle is the collection of all points so that the segment O, P is congruent to O, R. In particular, circles are always non-empty because the point R is on the circle. The center of course is not on the circle but it is in the interior of the circle, of course. The interior of the circle can be these points over here. If we wanted to put the circle and its interior together, we would refer to this as a disk, sometimes a ball is what it's called. So this would include the boundary and the interior. Exterior of a circle can be defined similarly. It's gonna be all those points so that the segment O, P is greater than the radial distance of O, R, of course. It's again, it's important to also emphasize here that the interior of a circle is not a subset of the circle because the circle is this curve right here. The stuff in the middle is not part of the circle. That's part of a disk. Important to distinguish between those vocabulary there. And so with this definition now in play here, I wanted to prove one proposition about circles to give us a flavor of how one works with circles in a congruence geometry. And then the other videos in lecture 21 will prove some other results about circles but I wanna prove one right here. So this proposition tells us that if a line intersects a circle, it happens at most at two points. And so that justifies the following type of drawing. If we have a circle, how can we have a line? Well, that line could not intersect the circle whatsoever, so to speak, they are parallel, the circle and the line. Most likely you're gonna get a line that cuts the circle into two different places but you also have the possibility of a tangent line. We'll define all of these properly on the next video here. And we also have the possibility that a tan line touches the circle at just one point. So this proposition is important to explain that there's no other possibility. You can't have a line that's like intersecting the circle in more than two locations. So something like this cannot happen with how lines and circles interact each other in a congruence geometry. So let's begin the proof then. Why does a line intersect a circle at most two points? Well, let's take a circle, gamma, and let's say that the center of the circle is O and its radius is OR. So then consider a line and we're gonna see what can happen here. So for the sake of contradiction, we're gonna suppose that the intersection between the line L and the circle, gamma, is three distinct points. And just as a curious little side note, when we hand write L, because we have to do a kind of cursive so it doesn't look like a one when we write L, but when we write gamma, right, depending on your font size, your L's and your gammas look like the same thing just upside down, that's kind of the intention I'm going for, we're using the same symbol for our generic line and generic circle just throw upside down, right? That's just a fun little thing in my perspective here. I don't know if that's why other people do it, but that's why I do it. But anyways, suppose that a circle intersects our line in three different locations, at least three, because there could be more than three, but there's at least three points of intersection. And so let's suppose we get something like that happening. So again, something weird could be happening like this. I'm not claiming there's other intersections beyond this three, but there's gonna be three points of intersection and because these three points are collinear, A, B, and C, after all, they're all on the line L, which L is illustrated right here on the screen. We can assume without the loss of generality that the point B is between A and C on this line. So we have A right here, we're gonna have C right here, and then B is between them. So something like that is what happens. And then when we introduce the, since we have the circle gamma right here, its center is O. And so each of these points, A, B, and C, are points on a circle. So they're radii, these segments, O, A, O, B, and O, C are all radii, and therefore they're all congruent to each other because they're all congruent to O, R itself. And this actually, believe it or not, forms some triangles. There is the triangle A, B, O, which is this triangle right here. And then there's also the triangle C, B, O, which is this triangle, excuse me, maybe this triangle right here. Now, these triangles we should mention, these are Asosceles triangles. Like if you look at A, O, B, like so, those two segments are congruent to each other. So by the Asosceles triangle theorem, we're gonna get that angle O, A, B is congruent to angle O, B, A. And likewise, when you look at the triangle O, C, B over here, that's also an Asosceles triangle, and therefore the angle O, B, C, which is this angle right here, O, B, C, it will be congruent to the angle O, C, B. I hope I said that right. O, B, C is congruent to O, C, B, which are those angles like so in these funky looking triangles, right? Now, we have, so we have these triangles. This is a common trick you do when you work with circles because all the radii have the same, the same, they're all congruent to each other. You can set up triangles in this situation, which are gonna be Asosceles triangles all the time. Now, since a triangle can have at most one acute, one non-acute angle in it, which means it could be a right angle or it could be an obtuse angle, okay? So be aware of that. That's what we mean by non-acute here. An obtuse angle is one that's larger than a right angle and acute angle is one less than a right angle. This was a consequence of the exterior angle theorem, which we proved previously in congruent geometry. A triangle in congruent geometry can have at most one non-acute angle. I want us to consider the two angles associated to vertex B right here. These angles are, this would be the angle OBA and the angle OBC. These angles are supplementary to each other. So if one of them was acute, like if this one was acute, that would force this one to be obtuse. And of course, on the other hand, if this one was obtuse, this would force that one to be acute clearly. So they can't both be obtuse. What you have here is one is acute, one is obtuse, or they're both right angles. That's a possibility. Now let's suppose that this angle right here was the acute one, which then would tell us this is an obtuse angle, but these two angles are actually congruent to each other. So you'd have two obtuse angles, which is a contradiction. Okay, well, let's go the other direction, right? Maybe this angle is the acute angle, which makes this one the obtuse one, but these triangles also, since they're exhaust, these have these congruent angles, this would be an obtuse angle. Oh, that's a JK there. So they neither can be acute or obtuse. So that means the only possibility is these are both right angles. So let us just put it this way. This is a right angle and this is a right angle. But again, these triangles right here would have to be a double right triangle, double right triangle. That can't happen in congruence geometry. So there's no possibility by the exterior angle theorem for this to happen. And therefore we get the contradiction that we were seeking that a line when it intersects a circle can be at most two points. Now, what we would want to follow up this theorem with is a proposition of the following form. Given three non-colonial points in our congruence geometry, we can form a unique circle that contains all three points. That's a horrible drone circle, but whatever. Something like circle determination, right? In incidence geometry, we have line determination. Two points determine a line. Well, do three points determine a circle? Well, clearly those points would have to be non-colonial. But is it possible that you could take three non-colonial points and always form a circle that holds those three points? That would be a lovely result, wouldn't it? And isn't that what we learned in geometry class? But unfortunately, such a statement of circle determination is not a theorem in congruence geometry. It's not even a theorem of neutral geometry. Like what was neutral geometry again? Neutral geometry is congruence geometry plus the continuity axioms. So you have things like the Archimedean principles, circular continuity, dedicating cuts. Well, these are all things we'll talk about in a future lecture for this series here. But even with continuity, which guarantees the existence of points that otherwise would be difficult to grab, even a neutral geometry's circle determination is not a theorem. And the issue is the following, hyperbolic geometry. In hyperbolic geometry, there do exist collections of three non-colonial points for which no circle contains those three points. Equivalent to this problem is we can't always inscribe a triangle into a circle. There do exist hyperbolic triangles, which cannot be inscribed into circles. This is something we'll explore in the future when we talk about hyperbolic geometry. But for the moment B, in other words, circle determination is independent of the neutral geometry axioms. Furthermore, we'll see later in our lecture series that circle determination is actually equivalent to the Euclidean parallel postulate in neutral geometry. So requiring that any three set of non-colonial points forms a circle is assuming the Euclidean parallel postulate, which is a very interesting thing you might not have expected. But as we're focusing just right now on congruent geometry, we do get that circle determination is not a theorem of congruent geometry. Hyperbolic geometry provides a counter example.