 Hi, I'm Zor. Welcome to Unisor Education. This lecture will be dedicated to introduction to triangles. Actually, it's about terminology, most of it at least, with very, very little proofs or theorems or anything like this. Triangles are pretty complex actually topic. So that's why I would like to dedicate this relatively short lecture to basically the terminology. All right, so what is triangle? It's a geometrical object which contains three different segments, and they are connected in such a way that the beginning of each segment corresponds to the end of the previous segment. And its beginning is corresponding to the previous segment, ending, etc., etc. So we have three vertices and three segments or sides of triangle. So these are sides, side, side, and side, and this is vertex. These are angles, obviously. So that's how it's taught. This is the definition of triangle. Now, what kind of triangles do we consider as separate classes, if you wish? Well, there are many different ways to classify the triangles. The first one is how the sides are compared. Well, if you have all three sides equal to each other, it's called equilateral, equilateral. Equilateral triangle means all sides are equal, okay? If only two sides are equal, then they are usually called either legs or maybe just sides, and the side which is not equal to both of those guys. So this is leg and this is leg is usually called the base. And if we are talking about vertex, it's not just any vertex of this particular triangle, usually it's the vertex which connects to equal legs. Unless specifically said that vertex at the base left or right or whatever. And this triangle is called isosceles, that's another word. All right, so that's how triangles can be differentiated or classified based on the lengths of their sides. Now, how about angles? Well, angles can be, as you know, right, acute, or obtuse. And that's why triangles with a right angle are called right triangle. This is called acute angle triangle or just acute. And this is called obtuse angle triangle or just obtuse triangle. So these are three categories which we based on the angles. Now, that's about it, about classification of triangles. Now, what's inside the triangle? What kind of elements of the triangle we will be considering? Well, besides vertices and sides, there are different lines or segments inside the triangle, which we will be dealing with. Now, first of all, let's consider a side and its middle point. And then the line which contains the opposite vertex with this middle point. So this is ABC and this is, let's say, M. AM congruent to MC. So these two segments have the same lengths. Now, the BM line is called medium. All right, so we know how to divide the side. Now, let's talk about the angle. What if instead of this, we will have these two angles the same. So BM is bisecting the angle. Well, if angle ABM is equal to angle to MBC, then BM is bisector or angle bisector. So these are different lines. Now, angle bisector might in some cases coincide with the medium. Well, sometimes not. It depends. And finally, if BM is perpendicular to AC, then BM is called altitude or height. So let's consider our triangle is such that all these three different lines do not coincide with each other. So let me use this and I will put... Okay, these are three lines. M, M, and P, let's say. So BM, in this case, if it divides AC into equal parts, BM will be medium. BM, in this particular case, let me correct this to the letter N. BM, angle bisector. So this angle is equal to this angle. So these are congruent angles. And finally, I will change M to P. So BP is perpendicular to AC. So BP is an altitude or... So this is the right angle. So these are three main components of the triangle. Now another thing which is sometimes also considered as a separate line. In this case, it's not a segment which is inside the triangle. It's a line actually. Line which is perpendicular to a side at its middle point. Now this is called midpoint bisector. So it bisects this side in two halves and it's perpendicular to it. Usually perpendicular midpoint bisector. That's how usually the full name of this line. It's a perpendicular midpoint bisector, if you wish. There are many different... there is no standard here. So these are important lines. Let's just leave aside midpoint bisector. I would like to talk about these three main components. Angle bisector, medium, and the altitude. What's important is that if you remember whenever I define something, I pay specific attention to two very very important components of a logical system which we are building. Existence of what I define and uniqueness. So I have defined let's say a bisector or a medium. Now, is it unique? Does it exist first of all? And if it does, is it unique? Well, let me just concentrate on medium first. I would like to prove actually that the medium does exist and it's unique. And it seems to me like an obvious fact. However, everything, even the obvious facts, need to be proven or taken as an axiom as you know. There is no axiom about medium being existing and being a unique triangle. There is no such axiom. So we somehow should derive this particular fact that medium exists and it's unique from certain axioms which we accept. Again, if you remember in one of the prior lectures, I was actually talking about Hilbert and his system of 21 axioms. And among them, we should really find something which could be used to prove this particular fact. Alright, so let's talk about medium. So first of all, let me say that there is a midpoint between two points on the line, between A and C, and it's unique. Well, existence of the midpoint is really a difficult topic and I don't want actually to get deeper into this. As I was saying before, the elementary things are most difficult to prove because there is nothing to be based upon. But its uniqueness actually is a relatively simple thing. Let's say you have a segment, in this case it's AC, and let's consider you have two different points, M and N, which both are midpoints of this segment. Now, what does it mean that they are midpoints? Well, AM has the length equal to output length of AM, equals to length of MC, since M is a midpoint. And similarly, I can say that length of AM is equal to length of MC, since N is a midpoint. Now I have to prove that M and N are exactly the same points. Well, let's consider they are different. So we will try to prove the theorem from the inverse statement and basically prove that the inverse statement is false. Well, let's have the lengths of these three small segments, from A to M, from M to N, and from M to C, as X, Y and Z. Now, what does it mean that length of AM is equal to length of MC? Well, AM is X. MC is a combination of M, N, and MC, so it's equal to Y plus Z. So that's what this particular equation means. Now, this equation means length of AM, which is X plus Y, equals MC, which is Z. Now, from these two equations, what can we actually derive? I would like to come up with some kind of contradiction that this is basically impossible. Well, if X, Y, and Z are, if Y is not zero. Well, let's try to do it. We can substitute Z into the first equation, and we will have X plus Y, sorry, X equals Y, plus, and instead of Z, we will substitute its value, X plus Y. Well, obviously, we can subtract X from both sides of this equation, and we will have zero on the left, and X and X will nullify each other, we will have two Y, from which Y is equal to zero. So, from these two things, we come up with Y equals to zero, which means that if we consider that these two points, M and M, are not exactly coinciding with each other, then we have the contradiction, then the length between them is equal to zero, which means that they have to really coincide. So this is a very elementary proof that there is one and only one midpoint of any segment. Now, how can it be used in this particular case? Well, obviously, since M, which is the midpoint of AC, unique, so it's just one point, now B is a given point in the triangle, and now we go back to the axioms, that if you have two points, there is one and only one line which contains them. That implies that the median is actually unique. Now, with angle bisector, it's actually exactly the same thing. So instead of caps of segments, I have to consider caps of the angle, which also have their own measures. So instead of lengths of segments, I should put measures of angles, and the proof is exactly the same thing. Now, how about height? Height is a little bit more complex. Why can't we have two different perpendicular from the same point outside of the line dropped on the line? Well, again, it goes back to the axioms. It goes actually, even simpler actually, you don't have to go through all the Hilbert axioms. You can go to Euclid's fifth postulate. So if you remember, the postulate has that if you have two lines crossed with another line, and if these angles, some of these angles is less than two right angles, then they have to really cross. And same thing on this side. These lines will cross wherever this sum is less than two straight angles. But in this case, we have two straight angles equal to each other, which means the sum of these two angles is equal to right angles, which means that these two lines cannot cross each other. So it actually corresponds to the fifth postulate of Euclid that you cannot really have two different lines dropped from the same point outside of the line on the line and perpendicular to the line. So that actually is how you prove that the altitude is unique in a triangle. It's actually very... I'm trying to emphasize once again that some elementary properties are more difficult to prove. You have to really know your axioms, what you can and what you cannot use. For instance, I cannot use, for instance, in this particular case, that the sum of angles in a triangle are supposed to be 180 degrees, because this is a much later theorem, which is proved after many other properties are proven. Because if I can use this, then obviously these two angles are right angles, so it's already 180 degrees, so there is nothing left for this angle, which means these two segments supposed to coincide with each other. But they cannot use this particular property because it would break the logical system of axiom, then the initial theorems, which are proven only based on the axioms, and then the second level etc. So we are building this building of logical conclusions only into one direction, only upwards, if you wish, from the axioms to more and more sophisticated theorems. We cannot use the theorem which is significantly above our level, wherever we are, to prove it. Because that would mean that we would have a logical loop, which is absolutely no-no in mathematics. Alright, what other interesting lines I missed? Well, there is one more, actually, I think. This is called mid-segment. The line, the segment which connects two middle points of two sides, it's called mid-segment. We will have certain problems related to properties of mid-segments. Well, basically, that's it. Again, it's a very introductory lecture about triangles, what kind of triangles exist and what elements we will be dealing with. There is basically a very descriptive notes for this particular lecture on the website. I do recommend to go to unicor.com not only to study these lectures, but also for parents, for instance, and supervisors to get engaged in supervising the educational process of their children and students, because they can actually use exams and tests. There are scores, so I do recommend actually to sign in as the real user of the system, not just a lecture watcher. That would help in a very important educational process. Alright, so in next lectures, I will start going into more details about triangles, but this is just an introduction. Thank you very much.