 Hi, I'm Zor. Welcome to Unisor Education. We continue talking about spheres in three-dimensional space. Certain theoretical considerations, I would like to present in form like a problem basically, so I do suggest you to go directly to Unisor.com to notes for this particular lecture and try to solve these problems yourself first. Doesn't really matter whether you will succeed or not. It's still useful to do it. You will put your mind into this kind of problematic mode. And another thing which is very important, my first problem which I would like to present is Well, it's again, it's more theoretical and it also involves very very fundamental principles in geometry and as usually the most fundamental principles is very difficult to prove because you really have to go to all the axioms and You don't have any other supporting statements, properties, characteristics, etc., to be based upon. So I what I'm saying in this particular case that whatever I present as a solution to problem one is not really a rigorous proof. It's more explanation of certain concepts in geometry. The rigorous proof is completely beyond the scope of this course. So, let me start. So the first problem is the following. If you have a sphere then I'm sure you understand that it divides an entire three-dimensional space into two subsets. The inside of a sphere and outside of a sphere. And the property of those points which are inside of the sphere is that their distance to a center is less than the radius of the sphere. Right, so let's say the radius is R. And the property of the points which are outside of the sphere is that they are further from the center than the radius R. It's obvious. I mean everybody understands it. But what if I will ask you, okay, how can you prove it? So my first problem is well, not to prove it, but at least give some reasonable explanation how we can approach the proof if given really lots and lots of ammunition to do it. Now well, first of all, obviously what makes sense to start with is to explain what is inside and what is outside of the sphere, right? Be intuitive and understand it. But what if I will ask you to define it in the more mathematical terms? Here is what I suggest and doesn't really I'm not really stating that this is the only definition, etc. etc. But here's what I would suggest. If you take a point which is outside how can you characterize that this point is actually outside? Well, let's connect it with the center. So my characteristic, which really differentiates inside from outside is that the segment which connects the center with our point, P, is supposed to intersect the sphere somewhere. If, however, my point, let's say Q, is inside the sphere, then this particular segment is not intersecting the sphere. So the characteristic which differentiates inside from outside is that the segment which connects our point with the center either intersects in case of outside or not intersect or doesn't intersect in case the point is inside. Well, that seems to be like, you know, good definition of inside and outside. Now, as a consequence of this, we see right now that in this case, I have three points on a segment. I know that this is R. This is the point where my segment intersects. So I know that this is the radius of a sphere. And since this point is in between this and this, this must be greater than R. Now, in this particular case, I know it's not intersecting, right? So let's extend it over and beyond the point Q, where it will intersect a sphere at some point N. In this case, situation is the following. This is R and this is Q. In this case, our point Q is in between O and N and O and N is equal to R. So it's supposed to be this distance supposed to be less than R. So these are considerations. And I would never call this a proof because, first of all, definition of inside and outside is really kind of questionable. This is something which I can suggest, but maybe there are some other definitions. And based on the definition, you really have to talk about the proof. So, but anyway, I think it's a reasonable explanation of what is this type of inside and outside concepts. And obviously, the consequence of this is that all the points which are outside are further than the radius from the center. And all the points which are inside are at the distance less than the radius. But that's my first problem. Now, my second problem is as follows. So if you have a circle, sorry, a sphere, and you have a plane which cuts this particular sphere at some kind of curve. So the plane intersects with a sphere in more than one point. That's important. Then, what's important is that if I drop a perpendicular from the center onto this plane, then this perpendicular would be inside a sphere. So whatever that curve is, doesn't really matter right now. I'm not really stating that this is a circle. That would be subject to the next problem. So far, I'm saying the only thing is that the point where my perpendicular falls onto this plane, the base of this perpendicular from the center of a sphere onto a plane which intersects my sphere. This point is inside. Now, let's go back to my previous problem. Inside actually is characterized by this distance to be smaller than the radius. So that's exactly what I'm going to do right now. And here is how. Let's just pick any point at the intersection of the sphere and the plane. And consider this triangle. Now, since point A belongs to a sphere, all A is equal to radius. Now, all P is perpendicular to the plane delta. And that's why all P is perpendicular to any line on this plane, including P A. So, all P A is at right triangle. Now, all P is a catatose, all A is hypotenuse. And we know that the catatose is smaller than the hypotenuse. And hypotenuses are, right? So the catatose, all P, is supposed to be less than R. And this is exactly the characteristic property of all the inside the sphere points. So the point P is supposed to be inside the sphere. That's my second problem. Now, my third problem is that this is actually a circle. So the intersection between a plane and a sphere is a circle. How can I prove that? Well, it's a flat figure since it belongs to a plane, right? The intersection. Now, let's take any other point here. P, B. And that would be another triangle. Let's think about these two triangles. All P A and all P B. They're all right triangles, obviously. They share the catatose, all P. And hypotenuses, all A and all B, both are radiuses of the sphere. So these two triangles are congruent, which means that the second pair of catatoses, they are congruent. So A P equals BP. So for any two points on this intersection, I have this equation, which means that any other point, C, D, or whatever, is exactly on the same radius. So P is a center of all the points which are equidistant from it, which is the definition of a circle. So whenever the plane is cutting the sphere, it cuts it along a sphere. That's very important. Now, this particular proof would not work if the plane goes through the point O, through the center of a sphere, because we cannot build triangles, the ones which I was just talking about. But this situation is actually even easier, because in this particular case, the intersection would be equator. And it's a flat because it's in the same plane. And all the points, since all the points belong to a sphere, all the points equidistant from all, and the distance is r. So basically, that's exactly what is supposed to be proven, that all these points are in the same distance from some point, which is a center. So this is supposed to be the intersection of the plane which goes through the point O, through the center of a sphere, is a circle. And usually it's called equator. But that's from Earth's geometry. Okay, that's it for this problem. And the fourth problem is, now I was talking about this particular plane, delta, intersecting along some curve. Basically, my point was that there are more than two points in an intersection. A and B are two different points. That's my proof was built upon. What if there is only one common point between a plane and a sphere? So it doesn't really cut it, it touches it. It's a tangential plane. So when the plane and the sphere have only one point of intersection, we are talking about tangential plane. So right now, I would like to prove that in this particular case, when the plane is tangential, then a radius, so this is the point where the point where the plane touches the sphere. So this particular radius into the point of touching is perpendicular to the plane, which is absolutely equivalent to a plane geometry. So whenever you have a line which is tangential line to a circle, the radius is perpendicular to the line. So this is a three-dimensional curve. Now, how can I prove it? Actually, it's very, very easy to do. Because think about this. If my plane touches only in one point P, everything else is supposed to be outside of the sphere. And we know that whatever is outside of the sphere has a distance greater than from the center, greater than the radius. So this OP is equal to the radius, and OA is greater than the radius wherever this other point A is. It doesn't really matter. Which means that OP is a shortest distance from the point O to the plane. So among all the points, OP is shorter than anything else. And that's basically a sufficient condition for perpendicularity. Because if, for instance, if we will consider that there is another point, let's say somewhere here, Q, and this is a perpendicular, so what happens? If this is a perpendicular, then this is supposed to be shorter than this one, right? Because this is the catheters, and this would be hypotenuse in this case. So OP would be shorter. But that's not the case. OQ is greater than R, and OP is equal to R. So Q cannot be any other point but point P. So that's basically the end of my small group of, I would say, very theoretical problems. I mean, something like the theorem about perpendicularity is really a theory. It's not really a problem which can be solved after all the theory has been learned. So I just presented all these little statements of theory of the spheres in the form of the problems. So I suggest you to go back to theunisor.com and review the notes for this lecture. And again, try to resurrect your own logic, your own proofs maybe, or these ones, whatever you have heard from this lecture. And these are always very nice exercises in just logical, strategical thinking about this. Well, basically, let me just repeat something which have repeated many times before. Mathematics is very rarely gives you some real knowledge which you can use in your practical life. What it gives you, it gives you an exercise for your mind, for your brain to be able to solve the real life problems. This just tunes up your thinking machine. That's it for today. Thank you very much and good luck.