 Hey, hello friends, welcome again to this session on James of Geometry and today We are going to discuss another very interesting theorem. It's called Morley's trisector theorem Now I have spent a good amount of time Constructing this diagram first before you, you know Get to understand what's Morley's trisector theorem. Let me just show you what went behind this construction So if I just show you this is the total construction, so don't get Bogged on by the construction which I have done It's Basically to trisect an angle and that to thrice that means there are three angles I have trisected all the three angles of the Given triangle, so let me just hide it so that it becomes clear to you So I have hidden everything which is you know the part of the construction which is not required for the proof and I have used the functions of hyperbola and Trisection of a line segment to trisect the angle that's another thing how I did it probably allowed to dedicate one full session on How to trisect a given angle Through construction, okay So now let me tell you what exactly is this theorem about so Morley's trisection trisection or trisector theorem says that ABC is a given triangle, okay? And you trisect all the three vertices a b and c so you can see I have trisected Angle a angle b and angle c the values of the angles have been given over here I hope you are able to see it for example here. It is twenty two point nine seven It is twenty one point five one at angle a and fifteen point five two degrees at angle b and What I've done is so you take the trisect, you know the point of intersection of the trisectors In an order for example the first one here and the first one here. They are intersecting here this point j one and The trisector this one this one here k1 b and k1 c is intersecting at k1 and Trisector C I one and a I one is intersecting at I one so these are the three points of intersection of the Trisectors taken one and you know at a time pair of them at a time and once you get this triangle Interestingly, this triangle is an equilateral triangle and that is what is both least trisector theorem so What is it once again? A triangle is there a bc you have trisectors drawn for each of the three angles and Let them intersect so the first trisector here is intersecting the Other trisector of the other angle at j one similarly k1 and I one and you join k1 j1 I one And it will always be a equilateral triangle. That is what was given by Frank Morley I think somewhere in Later part of 19th century So the proof is also very interesting, but this session in this session We are only going to validate it now the validation might look a little ugly because you know at times the moment I change the positions of B and C The figures might get degenerate points and hence, you know because hyperbolas and other you know Tools have been used. Can you see that? there are Three pairs of hyperbolas, which I have not drawn so it might so happen that You know some points are missing. So hence you can see some We are diagrams, but let us try anyways to validate it. So what I'm going to do is I'm going to change the point B and C, right? So these are variable points and let's see what happens. So let me stick this point B Yeah, you can see the points are missing at times, but then never mind. We will get something. Okay, so here Yeah, I got oh, I had just caught one over there So difficult to find out one. Yeah, here. Oh, I just missed it once again So you can see yes, this is another position where I'm getting this you know see the the construction of trisector of an angle was really a tedious job and You know if you go through the construction you will realize that it might so happen that The points of intersection which is there in the background are not real point and hence you see some kind of Weird figures in the middle, but then again, I have drawn another triangle. So this is another position where ABC is a triangle with all the you know, the Trisectors there so 23.96 18.35 and 17.68 these are three angles or in a Trisected angles that is and again, you can see this is 60 degrees The angles made here is 60 degrees. Let me change points see a bit and see what happens So let me take it to yeah, I had got one. Yeah, I'm intermittent points You can see lots of Yes, once again, I got another one So this is also where the three trisectors are intersecting and you get a equilateral triangle. So So now in this construction is becoming a little difficult to show you that every time you change the orientation of the triangle You will get an equilateral triangle because of the construction which I had Done right. So if I have a very, you know, or let's say a better way of trisecting an angle, maybe Then you will see a smooth transition And you can see a lot of processing is anyways happening the moment I'm changing one point So you can see what's there Okay behind so this is what the construction was like was you know, I have Kind of put everything in the background. So you got to see only this one Correct. So in the next session, we'll try to prove this theorem that will be really interesting proof So in this session, I wanted to just show you That what more least trisector theorem is now, I hope you understood it So there are three trisectors two trisectors each of each vertex angle Of the triangle and then they meet at three points. The three points when joined form an equilateral triangle always, okay, so this proof and a little bit of More on the properties of This particular theorem In terms of you'll see that few points lie on the circle, which all points I will explain The proof of more least trisector theorem. So see you again. Thank you