 Hello friends and welcome to another session on Gyms of Geometry. So today we are going to continue with the theorem which we proved in the last session. So if you have not seen that, I would recommend you just go and check it out once. So the previous theorem was that if I have a triangle ABC something like that ABC. I am just giving you a recap but you can just have a look in the previous video you will get the same thing. And there are triangles drawn like this. So let me name first ABC and there are triangles drawn on these exterior of the ABC triangle on the sides like that. So three triangles are being drawn let us say like that and let us say like this. Okay. Three triangles are drawn. Let us say PQR, PQR such that angle P that is alpha plus Q that is beta let us say and R let us say gamma. So if what did we prove in the previous case alpha plus beta plus gamma is equal to 180 degrees. Let us say this condition is fulfilled. Then the circum circle of these three triangles which one AQC. So let us say that is the part of the circum circle here like that and that of BCR. So let that be right like that something like that and AEB. So all three circum circles are going to have a common point like that. So I am just drawing roughly. So this will be the common point where all the three semicircles which one. So let me change the color for different triangles. So let us say this is one, this is one, let me, this is one right. The other one is let us say this one. So let me change the color. So this one. Okay. This is what we learned in the previous session. And the third one is let us say something like that, something like that right. So they will have a common point here P. Okay this was the case. So what is the condition? Condition is alpha plus beta plus gamma must be 180 degrees then it is true and we demonstrated, we constructed as well as proved it. Now this particular theorem which is called pivot theorem which was discovered in the year 1838. So this is special case of the theorem which we discussed in the previous session. So hence proving will not take much of a time here. So if we consider that this particular theorem which we discussed in the previous theorem is true then pivot theorem becomes very easy to understand. Now what is pivot theorem? Pivot theorem says that if points D, E and F lie on the sides AB. So this is AB, BC and CD, CA of a triangle ABC then the circum circles of triangle ADF, BED and CFE, these three semicircles, sorry, circum circles are going to have a common point which is P here. You can see that right. Once again this is called the pivot theorem where point P is called the pivot. So this is pivot guys, pivot okay, P is the point pivot. And what is this? It is saying points D, E, F lie on the sides AB, BC. So it is D is on AB, E is on BC and F is on CA of a triangle ABC then the circum circles of triangle ADF. So I have drawn this circum circle, this one is the circum circle of ADF. Circum circle of BED will be this and circum circle of CFE is this right. So they are indeed meeting at one point okay. And it has to be true. Why? Because if you look at this particular case, so let us try to relate these two cases. So here instead of ABC, the internal triangle which is ABC here I have demonstrated or showed it as BEF right. And the remote angles are AB and C in this case if you see. So there is a triangle made out of DF as a side and BE as a side and FE as a side on the external side of the triangle DEF. And very clearly A plus B plus C has to be 180 degrees. There is no doubt about it because angle some property if you are trying to. So hence by the theorem which we proved in the previous session we can say that the three same circum circles are going to meet at point P okay. This is called pivot theorem. Now let us try to validate it. So what I am going to do is in this construction I have shown three you know the point P ABCDEF and all. So let us try to change the locations of all these points which one. Let us say let us keep ABC as fixed and let us change the position of DF and E. So let us say if I take F somewhat here so you can see it is indeed in all the cases you will see that the three circum circles are meeting at point P. So let me just animate it fully. So now you can see a beautiful pattern is it? So can you see there are different locations of DE and F as DE and F are changing position. So is the triangle DEF and in all the cases you can see the circum circles are always intersecting at point P right. So the circum circles are always intersecting at point P that is there is a common point for the three same circles. So that is this is what I wanted to show you in this particular session pivot theorem the proof is not required or indeed or in fact we did the proofing because in the last session we proved a general theorem whose specific case is pivot theorem right. As I told you this was discovered in the year 1838 and the point P is called pivot right. So I understood this theorem. So let us meet with another I will meet all of you with another theorem in the next session. Thank you for watching this video.