 Hello friends So we have another theorem here and it says each angle bisector of a triangle divides the opposite side Into the segment's proportional length to the adjacent side Now you would have come across this Theorem already but in this particular session, we are going to prove it using trigonometry So, you know how to normally makes life easier Otherwise you can use the concept of similar triangles and basic proportionality theorem to prove it But we'll be using trigonometry and you know in a couple of steps to itself. We will be able to prove this theorem. So before we start, let's explain what's given. So, you know, this is what is given so ad ad is the angle angle bisector angle bisector of angle bisector Okay, angle bisector. Now you have to prove to prove. What do we need to prove? We need to prove ab upon ac, ab upon ac is equal to bd upon cd. That's what we have to prove it. Okay, and you will be using a sign law. So hence you can go to the first session where we have discussed the law of signs and using that it becomes very very simple proof. So, I can write in triangle abd in triangle abd using sign law rule. We can say ab by and let me call this as phi So this becomes 180 minus phi. Is it it? So ab upon sin phi ab upon sin phi will be equal to bd upon bd upon sin theta Is it it? Similarly similarly or rather let's first complete this. So this means ab by bd is equal to sin phi by sin theta clearly. Now in triangle, let's say a dc So we get ac upon sin of 180 degrees minus phi is equal to cd upon sin theta. Same using sin law. Okay, so hence from here we can say ac by cd is equal to sin 180 minus phi divided by sin theta. Now, we know that sin of 180 minus phi will be equal to sin of phi and this is sin theta. So from 1 n2 we learn from 1 and 2 we can say ab by bd is equal to ac by cd and hence rearranging will get ab by ac is equal to bd by cd. So the bisector ad divides bc in such a ratio such that it is in the ratio of ab by bc and it's proved