 Hi, I'm Zor. Welcome to Unisore Education. Today's lecture will be about circles, and I will start it with a bunch of definitions. It's really boring because there are lots of different elements of a circle, which we just have to know how they're called. So I will do this boring stuff as fast as I can. First of all, the definition of a circle. Relatively rigorous definition is, a circle is an object on a plane which contains points equidistant from one chosen point called a center of a circle. If you wish, you can use the word locus. It's a locus of points on a plane equidistant from one single point called a center. So any point on a circle has the distance to its center exactly the same. So all these segments which connect different points on a circle with the center, they all have the same lengths. So this is the definition of a circle. Now, circle has different elements, and I will try to draw them and define them. Circle, center. First of all, a segment which connects two points on a circle. It's called a chord. Chord, next. One particular chord which is crossing the center, so it's also a chord because it connects two different points on the circle. But if it also passes the center of a square, it's called a diameter. Next. Any line which intersects a circle in two points is called a second. The line which has only one common point with the circle called tangent. Now, if you have two points on a circle, then the part of a circle between these points is called an arc. Now, this is an arc and this is another arc. So basically two points on the circle define two different arcs, which complement each other to the full circle. Now, a part of a circle, let's use this two points again, and we have two radiuses into these points. This is called a sector. It's like a piece of pie, a piece of piece. What else? Segment. The part between an arc and a chord is called a segment. Do not confuse this area. Let me just put it this way, which is called a segment with a piece of a straight line between two points which is also called segment. Well, sorry, but this is the same word used in two different senses. So this is a piece of a plane if you wish, which is bounded by an arc and the chord of a particular sector. Sector is also a piece of a plane which is bounded by an arc and two radiuses. All other elements are linear elements or points. So we have points, we have linear elements like chord, diameter, second, tangent, and we have aerial, if you wish, pieces, aerial objects, which are part of the plane, which are sector and segment. Okay. What else do we have? Okay. Central angle. Okay. Angle between two radiuses, this angle is called central angle. Well, obviously, it's central because its vertex is a central square. Now, if however you have an angle, well, I don't have any space on this particular, let me write down. Okay. Here is another circle. Okay. An angle which is formed by two chords with the common vertex is called inscribed. Inscribed. Inscribed. Oops. My spelling, sorry. Inscribed angle. And finally, if you have a polygon with all vertices, a circle is called inscribed. These are all the definitions which I wanted to make. Too many different objects which we have to really just learn about. Okay. Another just small comment about radius. Oh, did I mention radius? Maybe I forgot to call it radius. Radius. So, radius is a segment which connects the center with any point. And now, since the circle is a locus of points equidistant from a center, then all radiuses have the same lengths, obviously. This radius, this radius, this radius, they all have the same lengths because they're all connected with the point on a circle. Now, the term radius we will use in two senses, actually, as the segment which connects center with a point on a circle, and as the lengths of this segment. So, we are talking about a circle of a certain radius. It actually means that the radius of the circle has certain lengths. Okay. That should not be really confusing at all. Now, any circle actually divides the whole plane into three areas which are not intersecting with each other. First area is all the points on the circle. These are all the points on the distance equal to the radius from the center. Then there are all the points inside the circle. These have the distance from the center less than the radius. So, equal to the radius, less than the radius. And finally, all the points outside of the circle are those points which have distance from the center greater than the radius. So, since these are completely different relations between two points, areas do not intersect. There is no single point which belongs to any two of these areas because these are usually contradicting conditions. Okay. Now, if you're simple microtheraums about circles, it's not even miniserums, it's microtheraums because the proof is actually like one sentence. Okay. Length of a diameter is equal to double radius. Okay. Now, this is a diameter which is a chord connecting two different points but it also goes through the center. Well, obviously, the lengths of the diameters equal to the lengths of these two segments and each one is equal to the radius. That's why diameter is always double radius in lengths. Length of a chord is smaller than sum of two radiuses. All right. Let's take a chord, let's say this one and two radiuses which we can draw to both ends of the chord. Now, there is an inequality of triangle, if you remember, that the sum of two sides of a triangle is always greater than the third side. Now, these two sides are two radiuses. That's why any chord cannot be greater than sum of two radiuses. And the only chord which is equal to the sum of two radiuses is a diameter. If two chords in a circle are congruent, then central angles are congruent. Okay. That requires a new drawing. So, let me just wipe out this to confuse us with all these little things. So, if we have a circle and we have two chords of equal lengths. Now, what this micro theorem says is that these two central angles must be congruent to each other. Well, obviously, since these chords are congruent and all radiuses are also congruent to each other, by definition of a circle, these two triangles are congruent and therefore, the central angles are congruent. Inverse is true. If the angles are congruent, then the chords are congruent as well. Well, again, it's very simple from... Equology from congruence of triangles. These two triangles will be congruent by side angles side because all sides are the same, they're radiuses. So, angles are given. That's why the triangles are congruent and that's why these third sides will also be congruent. Distance between any two points on or inside a circle is limited by a sum of two radiuses. Okay. So, if you have two points on or inside a circle, this is the center. So, the distance between these two points is always less than or equal than two radiuses, y. Since these two points are either inside or on the circle, then the distance, let's call it x and y, both x and y would be less than or equal than r. By definition of the concept of inside or on a circle. Remember, on the circle is when these distances are equal to r. Inside a circle is when the distance to a center is less than r. So, inside or on a circle, on a circle point, always has the distance from a center less than or equal to r. Now, since this distance between these two points by inequality of triangle inequality is less than sum of x plus y, that's why you have that x plus y is always less than or equal to two radiuses. And that's why the distance between these two points since it's always less than x plus y will be less than two r. Okay, a straight line as any other unbounded geometrical figure cannot be entirely inside the circle. Okay, actually, let me just have a small comment about this. What does it actually mean that the distance between two points is bounded by double radius? It means it's a circle is a bounded figure, which means there is no unbounded geometrical object which can lie completely inside it. And that's what the last theorem actually is. So, if you have an unbounded figure, let's say a straight line, it cannot lie completely inside because on a straight line, you can always find two points which have the distance between them greater than double r, obviously. And again, that's the definition of unbounded geometrical figure. Unbounded means that there are always two points with the distance between them greater than any number. That's what unbounded actually means. So, any unbounded geometrical figure, like a straight line, for instance, cannot lie completely inside the circle. Okay, now, it's intuitively obvious, but it's not so easy to prove this very interesting theorem. If you take two points, one inside and one outside of a circle, inside again means the distance is less than radius and outside means the distance to the center is greater than radius. So, if you have two points, one inside, one outside, you cannot connect them with any line which doesn't cross a circle. Again, intuitively, it's obvious. It's not really very easy to prove because what if, for instance, we have some holes here? Who knows? And we can always connect it through this hole, right? But anyway, it can be proven that the circle is a complete geometrical figure in the respect that this connection between inside and outside is impossible without crossing a circle. Now, we will talk about three points and a circle which contains them. First of all, let's take three points which are lying on the same line. My question is, is there a circle which would contain these three points? Well, again, intuitive answer is no, but it's actually quite easy to prove, and here is how. Now, let's consider a center. Let's assume that this particular circle does exist. So, let's have this center of this circle. So, the distance between these three points must be the same, right? That's the definition of points lying in the same circle. Now, if these distances are the same, then this center must lie on a perpendicular bisector of this sector, right? Because this is a locals of all points equidistant from these two. Now, at the same time, the same center should lie on the perpendicular bisector of this segment, so it should be somewhere here. Now, but we know that since it's on the same line, then these two perpendicular bisectors, they're perpendicular to the same line, which means they're parallel, which means they never cross. And that's actually the proof that these three points which are lying on the same line cannot be connected by a circle because the center must be on a crossing of two parallel lines, which does not exist. Now, what if they're not on the same line? Well, then actually we can always draw a circle around them. And here's how. So, let's consider our three points lying not on the same line. And we will use exactly the same logic since the center of a circle which is circumscribing these three points must be on equal distance from this and from this points. It must lie on the perpendicular bisector of this segment. Similarly, because the distance between this and this point from a center should be the same, then the center should lie on a perpendicular bisector of this segment, right? So, we have one particular segment between the points and another segment between the points. And we have a requirement that potential center of a circle which circumscribing these three points must lie on the perpendicular bisector of this segment and perpendicular bisector of this segment, which means it should lie on intersection of these two perpendicular bisectors. And since these lines are not parallel to each other, they always cross and there is always one and only one point which actually satisfies the requirement of being on equal distance from this and this, from this and this. And that's why the distance from all three points is exactly the same because this is equal to this because it lies on this perpendicular bisector. This is equal to this because it lies on this perpendicular bisector. And two lines can cross in one and only one point and this is a center of a circle and this is the radius of this particular center. And, by the way, we have also proven that if you have a third segment, then its perpendicular bisector must fall into this point which we have already received as a crossing of other two perpendicular bisectors. Now, why this perpendicular bisector crosses others in exactly the same point which they have already crossed? Well, it's very easy actually to demonstrate. If you consider that this is not the case, let's say, how can I draw it? I have to draw it incorrectly. That's difficult. All right, so let's consider our perpendicular bisector goes this way, perpendicular bisector of this segment. So it doesn't cross other points in this. So what do we have right now? Well, we have a very interesting moment actually that the intersection of this perpendicular bisector and this, let's say, consider this point. It also is equidistant from all three because it equidistant from this and this because it lies in this perpendicular bisector. It's on the same distance from this and this because it lies on this perpendicular bisector. So basically, if you wish, we have three different points in this case because any one of those points can be considered a center of a circle. So now, why is this impossible? Well, it's actually impossible because the points of intersection of, well, there is only one point of intersection between two different lines, which are not parallel to each other. And then, if we will assume, I think I have to have a better picture here, how can I assume something like this? All right, so it crosses this perpendicular bisector in this point and it crosses this perpendicular in this point and these two are crossed in this particular point. So let me wipe out these radiuses. So we will see exactly that we have three different intersections. OK, so what do we have now? We have now the following thing that since this perpendicular bisector intersects this line in this point in this point, it looks like we have this distance equal to this distance because they are all centers of the same circumscribing circle. But now what we have here is that if you remember, we had a theorem that the shortest distance from a line to a point which is not on this line is along the perpendicular to this. And the further you are from this point where the perpendicular falls, the longer the line becomes. It goes through this theorem during our previous lectures. So this is exactly the case because now this line and this line both are from a potential center to a particular point. They must have exactly the same lengths which is impossible because the further you are, the longer you are. So that's basically the proof of the fact that all three perpendicular bisectors of a triangle because we can talk about triangle right now since these are three points not on the same line. So all these perpendicular bisectors are crossing to the same point and this point is a center of a circle which is circumscribing this triangle. OK. Next. Next is we can use the fact which we have just established to solve the following problem. What if you have a circle and you don't know where the center is? How to find a center? Well, very easy. You just pick any three points and have this triangle and do perpendicular bisectors. They're all crossing to the same point which is a center. OK. That's easy. Next. OK. Next is about tangent. Now, you were already talking just a second ago about the fact that if you have a line and the point outside, then the shortest distance from the point to a line is actually a perpendicular. Everything else is longer. All right. Now, what if you have a circle and a tangent? Tangent is the line which has only one common point. Now, what I would like actually to prove is the following. First of all, there is only one point with the circle. Now, as a consequence, we can assume that there is no point inside a circle which lies on a tangent. Why? Well, because if you have a point inside a circle and you already know that there is a point on a circle, then inevitably it's a second which is crossing in two points the same circle. But our line has only one point in common. So there is no points inside a circle which lies on this line. All points are outside of a circle and only one on the circle. Now, outside of the circle are the points which have the distance greater than the radius. Now, and this point is on the circle which means its distance from a center is equal to the radius. What does it mean? It means this is a minimum distance from a point to a line which is a perpendicular as we were talking before. So basically it means that the radius which goes into this point where the common point between the tangent and the circle, this radius is perpendicular to a tangent. OK, so that's very important. Next, the reason inverse theorem. Here is the inverse theorem. If you have a line and you have a perpendicular to another line, which means first we have a radius, then we have this point where the radius actually is touching a circle and draw a perpendicular line here. Then this perpendicular line is a tangent, which means it has only one this point common with a circle, nothing more. It actually follows from the same theorem because since this line is perpendicular to the radius in this point, so this point has the distance from a center equal to the radius. Now, we know that this is a perpendicular which means every other is not a perpendicular and it must be longer than perpendicular, which means it cannot be on the same distance R from a center as this point, must be further. So all other points have distance greater than the radius, which means they are outside of a circle. Next theorem. If you have a radius perpendicular to a chord, it divides it in half. So this is a perpendicular. Now, why is this true? Well, again, remember that these are two radiuses, which means they are equal in length. So it's equilateral triangle. And in the equilateral triangle, an altitude, and this is an altitude to the side, which is not a pair of congruent legs of a triangle. This altitude is always a median and an angle bisector, if you remember. Since it's a median, these two are equal to each other. Now, inverse theorem is also true. If a radius is dividing a chord in two equal parts, it must be perpendicular. Same thing, actually. This is an equilateral triangle. Since it divides in half, it's a median. And we know that the median is also an altitude of equilateral triangle. All right. And here is the last theorem I wanted to address today. A second cannot intersect a circle in more than two points. So if you have a circle and a line which is crossing it in two points, the theorem says it cannot cross it in any other, the third point, so to speak. Now, how can we prove it? Here is the proof. Let's consider that there is a third point, which is also common between this line, second, and a circle. So this point, this point, and this point are all belong to both circle and a second line. Now, what does it mean? Well, it means that all these three distances are equal to each other. But now, let's forget about a circle for a second and consider, again, line at point perpendicular, which is the shortest distance. And then, again, the theorem that the further we are from this point where the perpendicular falls, the longer this becomes. So what do we have in this particular case? We have three different non-perpendicular lines which connect the point with the line, three perpendicular segments, so to speak. They're all having exactly the same lengths. Now, how is it possible? Well, it's not, because, again, as I was saying, the further we are, the longer this segment becomes. And the only way to have the two different segments is if they are on two different sides from the perpendicular, on the left and on the right, so to speak. Then these two segments are equal in lengths. There is no way you can have a third segment left or right, because here it will be longer than this one and here it will be longer than this one, or shorter, but definitely not equal. So we're only two equal in size segments which can connect the point outside a line with the lines. Well, basically, that's what it is all about. Thank you very much, and this is the end of this lecture.