 Okay, so we are going to discuss another theorem in this session. So this theorem is a triangle is divided by It's medians into six smaller triangles of equal area. So hence we know what a median is So median is a CVN CVN which divides Or rather which bisects Which bisects the opposite side, right? Opposite side that is a a dash in this figure is a median similarly BB dash and CC dash are median So they bisect the side BC CA AB respectively, okay So we have to prove that there all the three medians if you can see there are six parts of this triangle You know formed by the medians. So let me name them first. So let us say this side this triangle is alpha and this is beta gamma delta Epsilon and let's say sigma Okay, so I am Just labeling all these six areas now guys if you see Alpha is equal to beta Why? Because it is nothing but area of triangle G B a dash Is equal to area of triangle G a dash C. Why? because equal base Equal base both of them have equal base what base? B a dash is equal to a dash C because a a dash is a median and Same height same height same same height. Why same height because both of them have the common vertex G Okay, so alpha is equal to beta similarly. So this is one. Let's say Similarly Similary you can say gamma is equal to data Delta gamma is equal to Delta and let's say this one is to and Epsilon is equal to Sigma Okay, this is let's say three. Okay. Also Also area of Triangle a d a dash is equal to area of triangle a a dash C why again equal base equal base and That is B a dash is equal to a dash C and same height and Same right, right? So this means what do we get we get alpha plus? Sigma plus epsilon is equal to Beta plus gamma plus Delta now you see alpha is equal to beta so I can strike off alpha and beta and Sigma plus epsilon so sigma is equal epsilon so I can write to sigma Is equal to 2 gamma 2 sigma 2 sigma is equal to 2 gamma or 2 sigma Is equal to? Yeah, 2 gamma. I can say 2 gamma. Why because gamma is equal to Delta from here 2 and epsilon is equal to sigma That means this means sigma is equal to gamma Sigma is equal to gamma similarly, you can prove that Alpha is equal to Sigma You can do that as well, right? Why because if you can you write these two This thing what that sigma plus Delta plus gamma will be equal to Sorry epsilon plus gamma plus Delta is sigma plus alpha plus Beta right same reason now in this alpha and sigma are same Okay, so hence and gamma and Delta Gamma and Delta so gamma and Delta are also same so you can write 2 gamma is equal to 2 alpha So hence from here you get alpha is equal to Delta as well So hence instead of sigma you can write alpha is equal to Delta, but it hardly, you know matters why because Now you know that Sigma is equal to gamma and alpha is equal to Delta So hence all of them are actually equal So if you see alpha is equal to gamma and gamma was equal to alpha is equal to Delta and Delta is equal to gamma So hence if you see carefully all becomes equal alpha was equal to beta and Alpha was equal to Delta so I can write alpha is equal to Delta and Delta was equal to gamma So I can write Delta is equal to gamma Correct and gamma was equal to sigma so I can write sigma and sigma was equal to epsilon And so all are equal, all are equal, right? So hence what do we conclude? So the three medians together divide the triangle into equal areas, six parts of equal area. Is it it? Now the same logic can be used to establish that the medians divide the two triangles or sorry divides themselves into two is to one ratio that we will prove in the next session.