 Hi, I'm Zor. Welcome to UNISOR education. This lecture is about another construction problems in geometry, construction of a triangle knowing certain its element. Now this particular lecture is part of the course, which is called mass plus and problems where I'm trying to introduce certain maybe new concepts and much more different problems. Problems not exactly of the kind which illustrates the theory, but completely different type, which is supposed to kind of encourage you to be creative to think about something, to find the solution which is not really presented in in the theoretical course, which is the prerequisite for this one. It's called mass for teens and the same website, by the way, unisor.com So these problems are supposed to cause you to think about these things, to be creative, to be analytical. Okay, now the website unisor.com contains, as I was saying, prerequisite course for this one, which is mass for teens, which basically covers high school and a little bit more level of mathematics. It also has physics for teens, if you are interested in relativity for all. That's for those who are, again, more inclined to physics and more contemporary theories. Okay, now the website is totally free. There are no advertisements, no strings attached. Sign-in is not really necessary. It's optional. It's actually needed for a supervisory study because the website has certain functionality and parents can control what exactly the educational process of their children is. Okay, so let's get back to this particular geometrical problem. By the way, when we are talking about triangles, I usually have certain standards. My vertices are usually called uppercase Latin letters. Angles, corresponding angles, are Greek letters which correspond to the name of the vertex. Sites are called with lowercase Latin letters corresponding to opposite sides in the triangle. In this particular case, I would like to construct a triangle using its three altitudes. HA, HB and HC. Obviously, HA is the one which goes on to the side A. HB goes to side B and HC goes to side C. So using these three altitudes, we have like three segments basically, you have to build an entire triangle and quite frankly, it's not obvious how to do it. Now at this particular point you can pause this video, if you're watching this on the video, and think about this yourself. Now at the same time, I can give you a very very small hint and then maybe you can just pause the lecture. The hint is that the area of the triangle is one half of side times the altitude which falls on this side, but at the same time it's B times HB and times C times HC. It's a hint. I'm using this basically in the solution which I'm offering. Okay, so now you can pause this particular video and think about it, and I will continue basically trying to analyze the problem and find an approach to solve it. So this is very important. From this I can actually have simpler. I don't need area, I just need the relationship between A, B and C and altitudes. From this we can actually express everything using only A. Namely, B is equal to A times H, A divided by H, B and C is equal to A times H, A divided by HC. So that's what basically we need. This is a relationship between the sides of this triangle. So A, B, C are sides, H, A, H, B and H, C are corresponding altitudes and in any triangle you have this particular relationship between them. I will use this. Now I will use it to build a triangle which is not exactly what I've done, but it will be similar to what I want. Now if I will be able to do it, I will know the angles because similar triangles have correspondingly congruent angles. So that would basically solve my problem. Okay, now how can I do it? Alright, so let's just take any segment X and I will calculate segment Y and segment Z. These are sides of another triangle which is similar to the one which I need. Now, how can I do it? Very simply, I just have to put that Y is equal to X times H, A divided by H, B and Z should be equal to X times H, A divided by H, C. Now, if I will be able to do this, then let's do it this way. For instance, X is equal to certain coefficient K times H. K we don't know, obviously. We know X and we don't know A. It's just an analysis. Assuming that we have built this triangle with these altitudes, now the sides will be A, B and C and that's why I'm just choosing any kind of segment X and I assume that there is certain coefficient of proportionality or scaling factor between X and A. Now, Y is equal to K times A times H, A divided by H, B. But look at this. It's B. So it's KB. Now, what is Z? Z is X, which is K, A times HA divided by H, C, which is K times C. So, what's happening here? X, Y and Z are proportional to A, B and C with the same scaling factor K. So this is my analysis. All I'm saying is that if I will choose any segment X and calculate Y according to this and Z according to this, then my triangle XYZ would be similar to triangle ABC. Now, the question is how to do this? Well, this is, I think it was addressed actually in the main course how to construct these type of proportional kind of things, but it's really very easy. Let's talk about this one. So let's take any angle, any angle. Put here HB, put here HA, put here X and draw the parallel lines. Obviously, these are proportional, similar triangles, since these are parallel lines, triangles are similar. And from this similarity, you go that, let's say, and this is Y. So Y divided by X should be equal to HA divided by HB, which is exactly what we need here. Now for this particular case, again, similar thing. This would be HC. This would be the same HA as before. This is the same X as before. Now we'll draw the parallel lines, and that would be Z. Z divided by X is equal to HA divided by HC. Z divided by X is HA divided by HC. So this is how you build Y and Z knowing X and knowing these HA, HB and HC. They are given. Now what happened before I was just talking. It's all so-called analysis. It's just like assume that we have built triangle with these altitudes and its size are A, B and C. Then we have come up with this particular strategy. Now at the end of this, I really have to just make a solution basically the building itself, the construction itself I have to do. So what is the way to construct? Okay. As I said, we start from any segment X. That's the first step. Next step, we construct Y and Z using these two things. Angle can be any way the way that we're saying, so it's all very general. So now you know X, Y and Z. And we do know that X, Y and Z triangle is similar because it's the same scaling factor similar to ABC. Okay, so we know that. That's a very good solution. I have not constructed triangle with these altitudes, but I do have a triangle. Let's call it X, Y and Z with X, Y and Z. Since I know X, Y and Z segments, I can build this triangle, which is called X, Y, Z. It's supposed to be capital letter Y. Now I know it's similar. Well, if it's similar to ABC, it means that the corresponding angles are equal. So I have to build this one, ABC, and I know this angle. And I know all the altitudes, right? So I know all the angles because they are borrowed from this triangle, which I have constructed knowing X, Y and Z. So I know angles and I know the height. The altitude. Well, now it's easy. So let's say in this particular case, this is B, this is A, and this is C, and this is HB. HB would be sufficient. So I have two parallel lines on a distance HB from each other. No problems, right? And this is, I basically skip the elementary things like, okay, how to build two parallel lines on a certain distance. This is all much more simple and very close to theoretical problems, which have been presented in a course, mass for genes in a prerecorded course. That's why all these easy things I'm bypassing. If you have problem, for instance, constructing two parallel lines in a certain distance, you have to really go back and re-examine the course mass for genes in geometry, where I present all these problems and solutions, obviously. So now I have two parallel lines, which means my points, choose any point A. Now, how do I find point B? Well, I know this angle, so I'll just put this angle. That would be alpha, beta, gamma, alpha, beta, gamma. So I know now the point B. How to get to C? Well, this angle is beta. So I put this beta and that's my C. I didn't even use, in this particular case, HA and HC, they were actually built into this thing and construction of X, Y and Z to make this new triangle similar to this one. So they're built into the whole algorithm. But right now for construction itself, when I have already passed constructing Y and Z, X, Y and Z, now we need only HA just to build these two parallel lines. And here is my triangle. It's obviously again similar to this one because all angles are the same and it has correspondingly needed HB, HC and HA, all the altitudes. So this is a construction. So what I did was, first I did, okay, assume that my triangle exists and it has sides, A, B and C. Then these equations are correct. Then I have built X, Y and Z, choosing any X and constructing Y and Z according to the same formulas. And that's how I came up with idea of this triangle. So this is the first thing to build this triangle, which is similar to our. It's possible to do because we know all these coefficients and we chose X, X, arbitrary. Now continuation of the construction is, okay, now let's just use what we have. What we have is angles. And since we have angles, now this simple construction is actually working without any problems. Now, I didn't mention it before, but all the problems which I'm presenting and actually every video which is on the website on Unisor.com has a site, it has a textual description, which is basically like a textbook. And what I suggest you is always before or after you wish to lecture, I suggest you to read these particular notes. And again, it's basically, it's not very, very short kind of things. No, it's a full text of material which I present in the video. So that's why I suggest you to read it. Now, in case of the problems like this, in many cases, I present only the problem and maybe a hint, but not a solution. Solution is on the video. And that in my personal view is supposed to encourage you to do it again just by yourself. So after you watch this lecture, go to the textual part. So you have to go to Unisor.com. The course is called Mass Plus and Problems. Go to Geometry. And this is the lecture which is called Geometry 03. So you go to this lecture and you will see what exactly needs to be constructed. I'm not sure, maybe there is no hint, but in any case, there is no solution. In some cases, in some cases there is a solution. I think in this particular case I do present the solution, but you can just don't read it. Do it yourself first. In some other cases, when maybe problems are a little bit easier, I don't present the solution in the text of the notes. So I basically leave it to you. And I do encourage you to do it yourself. That's actually the whole purpose of the course, not just to listen to whatever I'm presenting or to read the solution if it's presented. It's supposed to actually encourage you to think. That's the most important part, because there is no practical implementation or whatever practical need to build a triangle using three altitudes. I mean, at least I just don't consider it. The problem and the purpose actually is for you to think about these problems and to find solutions. If you can find solutions in these strictly abstract kind of things, you will be able to find solutions in any practical problems as well. Practical problems also need solutions, which might not be provided for you, and you will have to find it. This is the practice. That's how you practice your muscle in the gym. You practice your constructing your creative abilities, solving the problems like this. Okay, that's it. Thank you very much and good luck.