 Hello friends, so welcome again to another session on gyms of geometry and today we are going to discuss another one very beautiful theorem and it's called steward's theorem and This was given by a Scottish mathematician Matthew Stewart in the year 1746 So this theorem is there since 1746 and you will later see that this theorem also is a general theorem for theorems like Apollonius theorem And which Apollonius theorem is usually taught in grade 10 Geometry curriculum now let's try to prove this theorem and before that let's first state what this theorem is So it's it talks about a triangle ABC where a x is a CVN You know what a CVN is a CVN is a point or sorry a line segment joining a vertex to the opposite side of a Triangle at any given point on the opposite side of the triangle Now it's given over here that the length of this CVN a x if you see in the diagram. It's P okay, and This CVN is dividing the Side BC into two parts one of length M and another of length N And you know that opposite side to vertex a will be small a Okay, and now it's the theorem says that a times p square plus mn within brackets Is equal to b square m plus c square n now before we start. I'll tell you a Small trick or no mnemonic so to say to remember this and it goes like this. So you can You know the mnemonic goes like this a man and his dad put a put a bomb in The sink a man and his dad put a bomb in the Sink so how does it you know relate to this one? So if you see a man and his dad, right? So hence if you see here this thing is Let's say, you know instead of P if it is let's say in the usual Diagram it is denoted as D. Let's say then what happens the same theorem becomes a D square plus mn is Equal to b square m plus c square n, right? So a man. So this is the man and his dad. So like that Put a bomb in a sink. So that is how you can remember this Theorem, right? This is a good mnemonic. Anyway, so let's try and prove this theorem So we will be using something called cosine rule in this case to prove this very very easily so what I'm doing is I am considering angle a x c to be theta and This is nothing but 180 degrees minus theta by linear pair, isn't it? So what can I say about B? So if you see if I write B square What is B square guys B square in our rather in triangle a x c In triangle a xc B square will be equal to D square Right instead of P. I can I'm now denoting it as D. So whichever way it is, you know, okay so D square plus n square and then minus minus 2 dn cosine of theta right and Similarly, if you write c square, so c square is d square plus M square minus 2 d times m times cosine of 180 degrees minus theta Now cos of cos or cosine of 180 degree minus theta is minus cos Theta, so hence this relationship becomes c square is equal to d square plus m square plus 2 dm cos of Theta, right? Let us say this is equation number one and let us say this is equation number two Now what I'm going to do is I'm going to multiply M into one and then I'm adding n into two Okay, so what would I get and the LHS you will get m B square plus nc square, isn't it? In the LHS and in the right-hand side you will get m times d square plus m times n square minus 2 d m n cos theta plus n times d square plus nm square plus 2 dm n cos Theta, so clearly this term and this term will get cancelled out. So in the RHS if you take Let's say what is common in this so if you see one m is common Right, so what yeah, so what I can do is I can take few common things here. So let's say I am writing Yeah, so let's say I'm taking m common in the first case. So you'll get m common d square plus mn and in the second case n common d square plus mn That means this is nothing but m plus n times d square plus mn and What is m plus n guys m plus n is a so simply a times d square plus Mn, so this is what the theorem is is it it so we could prove this now as I told you There is a special case and that's called Apollonius theorem. What is that? So the same theorem becomes Apollonius theorem Yeah, Apollonius theorem. How do you spell it? It is a p o and Wl o n i us Apollonius theorem Okay, so what is this so in this case in the Apollonius case Apollonius theorems case The CV and D becomes median so D is the is equal to a median Okay, then what will happen? This will make this will mean m is equal to n is equal to a by 2 Simply so hence if you see what will this theorem become so hence you can write a D square by 2 plus a by 2 c square is equal to a Times a square by 4 because D. Sorry not. Sorry. Sorry. D is D Yeah, so it is D square plus mn is a by 2 into a by 2 Is it let me just write it here now, right? So hence what will it become this will be equal to? So if you see in the LHS again, it will be um a by 2 B square plus c square and in the RHS Uh, what is it? It is a times Uh, so a one a will get cancelled so hence let it be like that. So D square plus a square by 4 Correct. So this a and this a will go so hence If I multiply the entire thing by 4 I will get 2 b square plus c square is equal to 4 d square Okay, plus a square or the same thing can be written now as D square plus c square is equal to 2 d square plus a square by 2 And again the same thing can be written as d square plus c square is equal to 2 times d square and Plus I can write a by 2 whole square Isn't it if you see this is same Now what is a by 2 a by t is nothing but half of bc So hence the same thing can be written in this form as well. This is called Apple on your serum. Okay, so this is uh one case of Stewart's theorem and what is the mnemonic a man and his dad a man and his dad Put a bomb in the sink Okay, this was given by steward in 1746