 Hi, I'm Zor. Welcome to a new Zor education. I would like to present to you a couple of very, very simple construction problems in solid geometry. Now, before talking about these problems, I would like to refresh your memory about certain fundamentals of construction in three-dimensional space. Because, as you know, we can't really easily draw things in this three-dimensional space. So, there are certain agreements between the people, certain considerations which we just take, and that's what I would like to talk before we approach the problems. So, first of all, very quick reminder of main axioms which we will be using. These are not complete set of axioms, but just those which we really need for these particular three-dimensional construction problems. So, the axiom number one, if there are two points on a line, line A, points M and N, which belong to certain plane. So, M and N belong to A, line A, and also they belong to plane mu. Then the whole line which basically connects them belongs to this plane. So, that's axiom one. Now, axiom two, if there is a point which belongs to an intersection between two different planes, mu and nu intersections. So, if planes intersect and there is a point M which belongs to this intersection, then they actually intersect across the whole line which actually passes through M. Well, graphically it can be presented this way. If this is one plane and this is another plane and this is point M which is in their intersection, then they intersect along the entire, so, exist line A which basically is their intersection. What are my, ah, mu and nu, mu intersect nu. So, if there is a point which belongs to intersection between two different planes, then there is a whole line which actually is their intersection. And obviously point M is part of this line A as well. So, that's the second axiom. And the third axiom is that if you have three points A, B and C not on one line, then there is a one and only one plane mu such that all of them belong to this plane mu. One and only one plane mu. So, these are axioms. We just take them without any proof. Not because it's really like an obvious thing. It's not really obvious, but we don't have a choice. Unfortunately, axioms is something which we don't have a choice but take it as a true statement without any proof because we cannot prove that because there is nothing more fundamental than these axioms. Okay, so these are three axioms. And then there are a couple of principles, as I was talking about, principles of construction in three-dimensional space. And these principles are if you have three points not on one line, then we can consider that there is a plane which is, which can be constructed. So, there is a plane which passes through all these three points. There is one and only one. That's an axiom. That axiom is, here is the difference. Axiom says that there is one and only one plane. Okay, as far as construction problems are concerned, we're saying that if there are three points, then we consider that the plane is also kind of given to us. We don't want to, you know, to draw anything in the space. We just consider that the plane is given if there are three points which do not belong to the same line but do belong to this plane. Now, another thing when the plane is considered to be given is if you have a line plus a point outside. So, if there is a line and point outside it, we consider that there is a plane which passes through the line and the point. Now, why? Well, very simple. Take two points on this line. Now, you have three points which do not belong to the same line, right? So, there is a plane which we can draw. Now, is this the plane which contains an entire line? Well, yes, because there is a first axiom. If you remember that if two points belong to a plane, then the entire line which connects them belongs to the plane. So, that's our second principle of construction. So, if we know that there is some kind of a line which belongs to a plane and the point outside it, we consider that the plane is given. Third one. If you have two intersecting lines, two lines and intersection not empty, we consider that there is a plane which contains both of these lines. Only one and we can consider that the plane is given if two lines are given. Now, why? Well, take this point, this point and this point. Now, you have three points which means you can have the plane, one and only one. And since these points belong to this plane, then the entire line belongs and since these two points belong to the plane, this line belongs. So, the plane actually contains both lines. So, again, two intersecting lines, that means that our plane is given. Two parallel lines, two parallel lines. Now, if two parallel lines actually have already proven that before, that if there are two parallel lines, then there is one and only one plane which passes through them. So, we consider that the plane is given as well. Okay. Now, if you have two planes with intersection not empty, two intersecting planes, we consider that if we have the planes, then we have that line of their intersection. We don't have to draw anything because the planes are not really drawn anywhere. But we consider that it, obviously, the theorem is true that there is one and only one line. Intersection, so. And finally, if there is a line and a plane and we know that the plane and line do have a common point, well, we consider if they're not parallel, we consider that the point is also given. So, there is a point. Point of intersection. So, anyway, these principles are considered as main principles in the construction problems. And now we will go through some construction problems. Okay. Consider you have a line A and you have a point B. You have a point M. And you have to draw a line B, which is parallel to A through M. Now, in the two-dimensional space, we know how to do it. However, now we have a three-dimensional space, which means we don't have any planes yet. However, let's resort back to the principles which we have. If there is a point and a line, we were talking about that we can consider that the plane, which passes through the point M and line A, is given to us. Okay. So, first, let's construct this plane, which contains the line A and the point M. Now, in this plane, we draw a line B parallel to A, and we know how to do it, because this is the plane geometry. And now, well, forget about the plane. The line exists on the plane, which means it exists in three-dimensional space. And this line is obviously parallel to this one. And why? Well, because of the definition of the parallelism between lines. Two lines are considered to be parallel in the three-dimensional space if they are in the same plane and they are in the same plane, because we started from drawing this plane, right? And they do not have any common points, and these do not have common points, because that's the parallel to A in the plane. So, that's basically it. Very simple thing. All we had to do is construct a plane, well, virtually construct a plane using certain principles and agreements, which we have. And within that plane, we resort to the plane geometry. Second problem. There is a plane, mu, and point outside it. Now we have to draw another plane, which contains point M, point, let's say, plane mu, and mu should be parallel to mu. So, construct a plane through a given point, which is parallel to a given plane. Okay, now we have to remember certain theorems in which we were talking about in the previous lecture when we were talking about parallel planes. Now, using one of those theorems, we do the following. We take any point on this plane, mu, given plane, and have two intersection lines. Now, from point M, in space, we don't have plane mu yet. So, in space, we can draw lines parallel to these ones. And we know how to do it, because it's just the previous problem which we just solved, right? How to draw a line parallel to a given line. Now, having these lines, and again one of the principles which we were talking about before, if you have two intersecting lines, we can always draw a plane which contains them. This is the plane mu. And basically my statement is that mu is parallel to M. And why? Again, one of the theorems which we were talking about in the previous lecture, if you have two planes and you have correspondingly parallel lines, then the planes are parallel. So, that's just back to the theorems which we have already learned. So, that's my second problem. Now, third, you have two lines in space. Let's say this is one plane, and this is another plane. Now, lines are not parallel and not intersecting. So, they are called skew lines. So, skew lines are like that. Non-parallel and no intersection. Okay, so, you have two skew lines. What you have to do is you have to draw... Now, these planes are not given. I just draw them for illustration purpose. Now, let's consider that you do have... Let's say this is E, and you have B. Okay, what you have to do is you have to draw a plane mu, this plane mu, in such a way that it's parallel to line B. So, again, I draw this for illustration purpose. There is none of this plane. So, we have only line B somewhere outside of this non-intersecting with A and non-parallel to A. So, we have to draw the plane mu. Now, how can we do it? Again, very simple thing. Draw a line parallel to B in this point. Now, these are intersecting lines, which means we can draw a plane, which contains these intersecting lines. Let's say this B prime. Now, why is mu parallel to B? Well, because B is parallel to a particular line on the plane mu. And one of the theorems which we have proved before was that if line parallel to one line on the plane, then it's parallel to an entire plane, which means no common points, basically. We did prove that. So, I'm always trying to refer you back to one of the theorems which we have already proven. Everything should be done hierarchically without any kind of a logical loop. So, whatever we have already proven, I am using. And nothing new. Okay, and the last problem which I would like to present today is, again, if you have two skew lines on this and you have a point outside them both. And you have to connect them with one line. To draw, you have to basically find two points in such a way that this is one straight line. The question is how to do that. Okay, here is how we will do it. Draw a plane which contains line B and M. Obviously, this line belongs to this plane. Why? Because this B belongs to the plane and point M belongs to the plane. Line and the point outside determine the whole plane. And since this line and including this point belongs to this plane, everything belongs to this plane. Now, completely different problem. Draw another plane which contains point M and A. Well, that's not a good drawing, obviously. It's supposed to be somewhere here. This plane contains A and M. Same thing, since A belongs to this plane and M belongs to this plane. Now, obviously, the entire line which connects one particular point of A with M also belongs to the plane. So, what happens in this particular case is that we have two different planes, one plane, let's call it mu, and another plane mu. And what's interesting is that this line which we are looking for should belong to these both planes. Because it belongs to this plane because it connects point M with one particular point on B. And if the continuation of this line connects to the A, then the plane which contains A and M also should contain the entire line. So, this entire line is contained in both planes, mu and mu, which means it's their line of intersection. And we were talking before that if there are two planes intersecting, then basically there is an intersection line between them, considered as a constructed, basically. Well, that was the last problem. Now, what's very, I would say, what's very beneficial for you would be if you would go through these problems again just by yourself and try to solve them yourself. Very, very educational, very beneficial for you. Strongly recommended. Okay, other than that, that's basically it for today. Thanks very much and good luck.