 Hello and welcome to another session on gems of geometry guys So in this session, we are going to discuss something called the Steiner lemus theorem The Steiner lemus theorem suggests that any triangle that has two equal angle by sectors Each measured from a vertex to the opposite side is isosceles meaning ABC is a given triangle where in BM and CN are the angle by sectors of Angle B and angle C respectively and it is given that BM is equal to CN and we have to prove that Angle B is equal to angle C that is or rather AB is equal to AC whichever way Now we will be using the theorem proved in the previous session To prove this particular Steiner lemus theorem and what was that theorem? It suggested that in a given triangle In a in a triangle and the previous session we discussed it if you have not covered that I would suggest Just go back and you know have a look on That particular theorem so it suggested that in a triangle in a triangle the bisector bisector of The smaller angle smaller angle is larger is Larger than the bisector by sector of the Greater angle. This is what we learned in the previous session, right? So what does it mean? It means that is there is a triangle Let's say there's a triangle like that okay, and This angle B. Let's say in this case. This is B and this is C A and B happens to be the larger angle here. So if the bisector of this Let's say BM will be Lesser than the bisector of angle C Okay, lesser than lesser than the bisector of angle C CN. Let's say right. So BM is less than CN if Angle B is greater than angle C This is what we learned in the previous session. Now that is so in this case as well Then the bisectors must be, you know, greater than or less than each other, isn't it? But here We have been given that BM Is equal to CN This is given this is given that means that means we can't really say which angle is Greater what I'm what I'm trying to say is if angle B Is greater than angle C. Let's say if it was the case then BM should be less than CN according to This particular theorem which we proved in the last case. So let's say case one Right, but this is not true because BM is less than CN not true Not valid Okay, case two If angle B if Angle B is less than angle C then Then what will happen? BM Will be greater than CN Correct By the same logic which we proved again this one this logic But this is also not valid because we know that the given thing is That BM is equal to CN. Hence the only possibility is case three That is angle B must be equal to angle C Okay, so this is the only possibility only Possibility Is it so hence we conclude that triangle ABC Is an isosceles Right hence proved So the major part of this proof actually came from the previous proof That is in the previous session whatever we have discussed If you go through that proof you will be able to clearly come out with this result, right? So please remember This is called the steiner lemus theorem now if the two angle bisectors are same equal in any given triangle You know for sure That the triangle is isosceles and you also know Which two sides are equal in that triangle?