 Hello friends and welcome to another session on Gems of Geometry and again as we have been doing so far, we will be discussing another very, very important, very interesting theorem. So what is this theorem and after this theorem or rather this theorem becomes the basis for many other theorems like for example what Brahma Gupta found out in terms of let us the area of a cyclic quadrilateral. So after this theorem we will see how there are three cyclic quadrilaterals which will be having the same area and that is how it becomes the basis for the very fact that the area of the cyclic quadrangles are a symmetric function of their sides. So this theorem which we are going to discuss will become the basic premise for that theorem which was actually given by Brahma Gupta. So let us now talk about this particular theorem here. So in this session what I am going to do is I am going to construct the entire theorem that means we will be constructing and we will be demonstrating through construction and the construction demonstration will be good enough to prove this theorem as well. So let us begin. So first let us understand what the theorem is saying. It says any four unequal lengths each less than the sum of other three will serve as the sides of three different cyclic quadrangles all having the same area. So let us deconstruct it. So basically what it is saying is if A, B, C and D are four lengths such that A plus B plus C plus sorry A plus B plus C is greater than D if this is true for any choice of A, B and C. So don't just think that I am taking A, B and C. So hence it should be A, B and C the first three only know. So any combination any three of these four lengths you pick up and if the sum is more than the fourth one then what does it say? These four segments can be of or can be laterals or sides of three different cyclic quadrilaterals and all will be having the same area. That is what we are going to establish here. I hope you understood. So basically let us say if I am drawing a rough sketch here. So let us say if this is A, B, C and D and let us say there is a circle which passes through all the vertices. All the vertices let us say. So in that case they are saying within the same circle with the same A, B, C and D we will be having three different cyclic quadrilaterals. How? Let us see that. So let me first clear this canvas. So now I am going to use this algebra tool to draw and demonstrate. So what I am going to do is first let me take a circle. So I am taking let us say this point here as the center and let me take a radius of 5. So this is the circle and let me rename this to O. So our standard center's name. So rename it to O. Here it is. Now let us take four points on this circle. So first point is let us say this one A. Second point is let us say this one D and let us say this is third point C and let us say this is D. So now all these three points or sorry four points are on the circle and now we are going to draw a quadrilateral or quadrangle. We will join all these. But while drawing what I am going to do is I am going to you know demonstrate that particular quadrangle as some of two triangles. For what reasons it will be clear little later. So let me draw one polygon or triangle A, B, C. So this is one. And then the other one is A, C, D. So you can clearly see that A, B, C, D becomes the cyclic quadrilateral in this case. And A, C is the diagonal. Now what I am going to do you have to pay a little bit attention here. So what I am going to do is I am going to reflect one of the two triangles. So you can see this cyclic quadrilateral is made up of two triangles A, B, C and A, C, D. Now I am going to first create a mirror which is perpendicular to this diagonal and then I am going to reflect the triangle A, C, D on that. What do I mean? Let me do that and show it to you. So first of all I am going to take up a perpendicular line. So how to draw a perpendicular line? I am just taking this arbitrary point. And from this point G on this line I am drawing a perpendicular. So this line the new line which has generated has been generated is perpendicular to the diagonal A, C. Why? Now I will be reflecting the triangle A, C, D about this line. So let me reflect that. So I am selecting reflect this part, this triangle and this is the mirror let us say. So here is the reflection. So do you see there is a reflection of A. So let me just zoom out. So A, C, D and A dash, C dash, D dash is the reflection of triangle A, C, D. Correct? So since it is a reflection don't you think that these two triangles which one A, C, D, A, C, D and A dash, C dash, D dash are congruent. So here are the simple reflections. So now what I am going to do is I am going to translate this triangle back to its original diagonal. So let's say I am coinciding A dash to C dash and A to C. Okay? Right? So don't you think the quadrilateral, there are two quadrangles now what all? So let me just clear the clutter here. So there are two points. So two points A and C dash are coincidence and coincident and here C and A dash are coincident. So the two triangle part is clear to you I believe. Let me just take this off. So this will not create any hindrance in understanding. Okay? So let me just zoom it in and yeah. So I think now it is clear. So what I have done is I have reflected one of the triangles about its you know the perpendicular line to the base. Okay? So you know that triangle A, B, A, D, C. A, D, C is reflection. So let me just highlight this. So this is A, B and C. Its reflection is A dash, D dash, C dash. So both of them are congruent anyways. So both of them are congruent that means their area is same. So don't you think that we have two cyclic quadrangles now which one? So I am writing them here. Let us say A, B, C and D dash or first A, B, C, D and A, B, C and D dash. Both are having same area. Right? Both are having same area. And in fact if this side was let's say A, this side was B, this side was C and this side was let's say D of the first quadrangle A, B, C, D. Now on the other side this one will be C and my friends this is D. D dash, A dash is D because of the congruence principle simply. So you can see there are four values of or rather there are four lengths A, B and C and D and they are constructing two cyclic quadrangles with the same area. Right? Why is the area same? Because the two quadrangles are made up of same constituent triangle. So hence areas are same. Right? So this is what we have obtained. So at least two quadrangles definitely are there which are having same set of A, B, C, D sides and which are having same area. Now what about the third one? So how do we get the third one? So what do we do? So let us first clear this you know whatever we have marked on the screen so that becomes easy for us to understand. You keep in mind what are A, B, C and D. Okay? So I have cleared it out. Okay? So these two quadrangles are equal area with the same sides that we have proved. Now what I am going to do is so earlier I took the diagonal A and C, AC. Right? Now I am going to take this diagonal or I am going to make another polygon look here. So B, C, D and B back. Okay? So this is another polygon and now let me switch off the first reflected one. So this was the first reflected one. So let me switch it off and I have made another diagonal like that and let me take another point again. So let me say this is another point and I am going to make a line perpendicular to this diagonal BD passing through H. Why? Because I am basically creating a mirror. Okay? So let me draw a perpendicular from H perpendicular to this line BD. Okay? Now that means I can reflect this triangle about this new mirror, so to say. Right? So what I am going to do is I am going to reflect this triangle about this one. Fantastic. So I got a reflection of B, C, D as B dash, C1 dash and D1 dash. Okay? So I hope this is clear and I am going to do the same thing. What am I going to do? I am going to shift or translate this triangle back to this circle. Right? So I am just fitting it in here. See? Perfect. Oh, but it actually was, you know, so the choice of point was not that great. So I have to change that because it is exactly matching. Yeah? So I am not getting two different things. So I have to just repeat it. So no worries. So what I will do is I will just delete this part. I don't need this one. So I have to basically change the, this is because of the configuration so that two triangles basically coincided. So hence what I am going to do is I am going to shift this point. Okay? Oh, let's say like that. Yeah? In that case, mostly I will be getting a different triangle altogether. Okay? Now what I am going to do is again, I am going to take a point, another point and from this point edge, I am going to drop a perpendicular on this line. Perfect. So I did that and I am going to take a reflection of any of the two triangles. So I can take either this one or that one. So let me take the bottom one. So I am going to reflect this one about this one. Perfect. And now I am going to move this back to, so let me move this triangle, new triangle which I generated, which is reflection, back to, yeah. Now this will make some sense. Okay? So here is that guys. Okay? So let me just declutter it a bit. Let me take it away. It is not required anymore. And these points, let's say this is C1 dash, this is C. So I hope this is clear. So how many quadrangles can you see? So let me just take it away as well. So you can see third quadrangle being generated. And I hope you understood that, you know, this will also be, this area will also be same. What area I am talking about? So let me open it up once again. So I am saying, okay, before that, let me just declutter it once more. Here are few points which are coincident. So let me take it here. So these are all same coincident points. This one also. Okay? Yes. Okay. Now what I am saying is, if you see again triangle B, C, D is congruent to triangle B dash, C dash 1 and B or D dash 1, whichever way you want to take because these are coincident points. Right? Look at the figure. Again, B, C, D. This is one triangle. And I reflected this triangle about that mirror if you remember. And then the reflected triangle was B, C1 dash and D. So these two, this one. So these two triangles are, so this one. So I am basically saying this triangle and this triangle and sorry for the bad drawing guys. So let me re-draw it once again. So let me take another color so that becomes easier. So this triangle, this one, this one, this one and the other one. Let me take this color now, this one. These are congruent, is it? These are congruent because they are reflection of the same triangle. Okay? So since they are congruent, then if I add this particular area to both of them, I will get the same area. And if you see, the sides are same. So if this was A, this was B, so this was C, let us say and this was D. So A and D stays the same. So in the next diagram or when we reflect it, this becomes B and this becomes C. Isn't it? So the sides A, B, C and D are still the same. And the area of the new quadrangle, which quadrangle? So A, B, C1 dash and D, area of this quadrangle is equal to area of which one? We can say B or A, B, C and D. That was the parent or the original quadrangle, isn't it? Now let us see all the three together. For that what I need to do is I need to just clear this off so that it becomes easier for you to see all of them together in one frame. Okay? So now I am going to switch on the triangle which I had switched off earlier. So let us see. So let me show the object. So this one, okay? So now all of them in the same frame. Can you see the three quadrangles? Three quadrangles. Clearly one is A, B, C, D, A, B, C, D, the parent one, first one. Then I had reflected this triangle A, C, D, isn't it? I had reflected A, C, D and got A, C, D dash, right? So the two, then let me write all the three quadrangles and you can view it and you can also analyze from your end. So I am saying area of quadrangle A, B, C, D, the original one is equal to area of quadrangle A, B, C, D dash because of the reflection thing. And the same thing is area of the same A, B, C, D from here, if you see. Here you see, you see. I can write that as A, B, C1 dash, D. Okay? So there are three quadrangles with the same area and also all of them had sides A, B, C and D, isn't it? So this was A and let us say the original one. So this was B and let us say this was C and this was D, the original, right? So the same A, B, C, D can be A is same. Now this B comes here because of reflection and also and this is C. So this is C, this one. Okay? And what about D? So if you look at it, D happens to be this one. So this side is also D and this side, this one, A and D dash is same as C. Okay? So you can see all the four quadrangles have A, B, C and D as the sides. Okay? A, B, C and D as the sides. So right? So sides are, need not be in the same order but sides are same as in, they have the similar segments or same segments in all the quadrangles as well as we proved that the area is also equal because for area we just, you know, cut the diagonal into two pieces along the diagonal, sorry, cut the quadrangle into two pieces along the diagonal and just reflected one of the parts by 180 degrees, right? And then so area has to be same and the sides are same. So this is what we wanted to prove that if there are four unequal lengths and each less than some of the other three, then all four of them can form three different cyclic quadrilateral or quadrangles with the same, within the same circle with the same area. Right? So now we have proved it. I hope you like this one. So let us now go a little bit more deeper into different theorems related to quadrangles and especially cyclic quadrangles. Thank you.