 Hello and welcome to another problem-solving session on triangles and we are solving problems related to internal angle bisector theorem So let's see and go through this particular problem So this says if the bisector of an angle of a triangle bisects the opposite side Prove that the triangle is isosceles. Okay, so let's first draw a triangle So let's say this is a triangle. Okay. Let us name this triangle Let us name this triangle. So a b C Okay, now, what does it say if the bisector of an angle? Let's say we draw a bisector of triangle angle a or B a c angle B a c. This is the bisector. Let's say Okay, so let me draw once again So yes, so this is my Bisector, what does this mean? This means that this is angle X so This one is also angle X Okay. Now what we have to prove that triangle is isosceles. Okay, let's see By sex the opposite is also given that this is D. Let's say so If that is D, then we know that BD is equal to DC. This is given. So Let's write given what is given AD is Angle bisector of Angle B a c correct This is given number one given two is BD is equal to DC BD is equal to DC. So in other words AD is the median Right AD is the median So the theorem could be given as if AD is a median as well as angle bisector of angle a then the Triangle is isosceles. Okay, so to prove To prove the triangle is isosceles meaning what you have to prove AB is equal to AC Right, let's see how to prove this. So the moment there is a angle bisector What do we Get a hint as so we get a hint as that internal angle bisector could be used over here, isn't it? so if that is true that means in triangle ABC What can you say? since AD is an Angle bisector internal angle bisector that is Hence We know that the ratio it will divide the opposite side in the ratio of the adjacent side, isn't it? So AB upon AC is Equal to BD upon DC. Isn't it? This is given but but since BD is equal to DC Right BD is equal to DC. So hence, let me solve it here now. Therefore AB by AC Will become one why because BD and DC are same so BD by DC will become one BD by DC is one so AB by AC is one therefore This will mean AB is equal to AC and that's what we needed to prove So it was very Simple problem. Correct. So again like learning is if you see there is a bisector involved Then and we are talking about ratios of the sides of the triangle then internal angle bisector comes handy to solve such Problems