 So, let's take up another theorem. In this theorem, it says, medians of a triangle trisect each other. What does the word trisect mean? Trisect means dividing into three parts. So for example, or basically the point P is dividing AB in such a way that AP upon AB is equal to 1 by 3, or AP upon PB is equal to 1 by 2. This is called point of trisection. Okay, so let's try to prove that medians trisect each other. What does it mean? So, let's say this point is O, the centroid is O, then we have to prove. First of all, the given is that AD, AD, BE and CF are medians, medians of triangle ABC. Okay, and you have to prove that to prove, to prove. What do we need to prove? We need to prove that AO by OD is equal to BO by OE is equal to CO, CO by OF, right? CO by OF all are 2 by 1. This is what we need to prove. How to prove it? So, we will use the theorem just proved in the previous session, we are in triangle, sorry, medians, medians of triangle divide it into six parts or six triangles, six parts or six triangles in this case, six parts of equal area. We just proved it in the previous session. You can always go back and check that session. Okay, so medians of a triangle divide it into six parts of equal area. So, if you see, what does it mean? Triangle OBD is equal to, or area of triangle OBD is equal to area of triangle ODC is equal to area of triangle OCE is equal to area of triangle OEA is equal to area of triangle OAF is equal to area of triangle OFB and all I have labeled as x all these areas are same okay let's talk about triangle a b o and triangle a b o and triangle OBD these two triangles so I can write area of triangle a b o divided by area of triangle OBD is nothing but let's say half into base base for the first case is OA into let's say H the height and similarly half into OD into H H being the let's say this one is H the perpendicular length is H can I do that yes I can so if you see the left hand side of this relationship a area of ABO is nothing but X plus X if you you know see clearly you see here closely sorry so if you see closely it's X plus X which is area of ABO let me just highlight it so if you see this is ABO X plus X and what is a OBD OBD is this much right so hence clearly I can say this is X plus X divided by X and hence it is 2 by 1 so what do we infer we infer that OA by OD is equal to 2 by 1 right similarly you can repeat the process for the other two pairs of triangles and hence you can say similarly OB by OE is equal to OC by OF and all are equal to by one and hence we could establish the given theorem right so hence medians of a triangle trisect each other is proved now you can use multiple other methods to prove this for example you can use coordinate geometry you can also use Pythagoras theorem multiple other theorems you know to prove this but this one looks very very simple okay so the medians of a triangle trisect each other