 Hi, I'm Zor. Welcome to Unizor Education. We will discuss today the tools of geometry. Well, obviously we are talking about virtual tools, not the real ones. Since everything in the real geometry is virtual, it's our imagination, it's in our fantasy, it's in our brain actually, which really exists. It's all abstract concepts which we are talking about, when we talk about points, aligns, circles, planes, etc. in geometry. So what are the tools which we can imaginary build all these elements? Well, there are two main tools. One is called the ruler and another is called the compass. So what can we do with these tools? Well, again, these tools are abstract, which means we are capable of doing certain abstract things with abstract objects. For instance, if you have two points on the plane, we are talking about a line which contains these two points, or a segment actually in between them. Now, how can we get this line? Well, using the ruler we can. So we are assuming that there is a ruler, no matter how far from each other these two points are located. We are talking about abstract points on abstract plane. So this instrument ruler is an abstract ruler. So it's infinitely precise as far as the straightness is concerned, and infinitely long because it's sufficient to draw a line which is basically infinite, or a segment of any particular size, no matter how far from each other these points are located. Now, another instrument is a compass, and the compass allows to draw a perfect, ideal, abstract circle around a particular point as a center using a particular segment as the radius. So this is ideal ruler and ideal compass. We can use obviously any radius or any lengths between two points, and we still are capable of doing something with these two tools. Now, why can't we use something else? Well, quite frankly I don't know, but it's a tradition in geometry to use only these two instruments to prove the theorems, to solve the problems, to construct certain geometrical figures, and whatever is not constructable or possible to prove using these two instruments is just considered to be outside of the scope of the geometry. And here are a couple of very interesting examples. We will definitely go through all the different construction problems in some further lectures down the line, but I just want to mention something. For instance, you have a problem like this, a very simple problem. You have a segment and you have to find the middle point of this segment. Well, there are many different ways of doing it using ruler and the compass only. We are only talking about the ways to do this with ruler and compass. And ruler, by the way, doesn't have any measurement bars or anything. Well, one of the things which you can do, for instance, and I'm not going to prove it, I'll just basically construct it without any proofs. You take the compass and have any radius big enough, let's say the whole segment is a radius, and you draw a circle. Let me just put a little smaller segment so it will be better. So you use this radius to have one circle using this as a center. And then using this point as a center, exactly the same radius, you have another circle. Now, if you connect with the ruler these two crossing points, then the crossing point here will be in the middle of this segment. Now, I'm not going to prove it, it will be further in some other lecture, but I just wanted to make sure that we can do something with these two tools. Unfortunately, there are certain problems which are not solvable using only these two tools, two instruments. And here is an example of something which was actually known to Euclid to be a problem which he did not really know how to solve. For instance, you have an angle, and you would like to trisect it, which means you have a couple of arrays from the same vertex in such a way that all these three angles are congruent to each other. Well, it's not possible to do with ruler and the compass. What's interesting is there were many people who basically talked about certain solutions, they offered certain solutions to this problem, which is called trisection of the angle, and roll rock, basically. It's actually proven that it's not possible to trisect any angle, better but any angle. It's not possible to trisect an angle using the ruler and the compass. No matter how powerful we consider mathematicians are nowadays, there are certain problems which are not solvable within the framework which they have imposed on themselves. Now, this is a framework. Mathematicians imposed on themselves this restriction, and using this framework, you cannot solve certain problems, and trisection of the angle is one of them. Another example of a problem which is also not solvable and again, it's proven that it's not solvable. It's not just mathematicians cannot come up with a solution. There is no solution. So, here is the problem. If you have a circle, how can you build a square which has exactly the same area? It doesn't matter what really a rigid definition of area is. Again, we will cover it in some other lecture, but people do understand generally speaking what area is. So, you cannot build a square which has the same area as a given circle. That's another example of a problem which is not solvable using the ruler and the compass. These two tools which are only possible, which are only agreed upon, rather, to be used in geometrical problems. So, anyway, in all the future, proofs of theorems, problems, constructions, etc., we will do only actions, or we will do any constructions, any drawings, or whatever, which can be done by ruler, without any marks on it, and a compass. That's it for today. Thank you very much.