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I've drawn an arbitrary triangle right over here and I've also drawn it's three medians median EB median FC and median ad and we know that where the three medians intersect at Point G right over here we call that the centroid what I want to do in this video is prove to you that the centroid is exactly 2/3 along the way of each median or another way to think about it we can pick any one of these medians and let's say let's pick a B what I want to do is I want to prove that eg eg is equal to 2 times G be 2 times G B so whatever distance this is it's twice this distance there or another way to think about it is eg is 2/3 along the way of e B and the logic that I'm using to prove this you can use for any of the medians to show that the centroid is exactly 2/3 along the way of any medians or divides it into a segment that's twice as long as the other segment and to do that let's focus I want to focus on triangle a B II triangle a B II right over here and I'm gonna draw this median as essentially the base so let me draw it that way I'm gonna try to color code it similarly so we draw it one draw a little bit flatter than that so it's like that and then we have the two yellow sides so it looks something like this it looks something like that and then we have the centroid right over here at G that is our centroid and then we had this magenta line going to a let me draw it a little bit neater than that we have that line going to a and then we have this blue line going to F right over here we have this blue line going to F and let me label all the points go back to the orange color so this is going to be e this is going to be B this is going to be a this is going to be F right over here and just to make sure we have all the same markings that little marking is that marking these two markings these two markings are on this side right over there and the whole way that I'm gonna prove that eg is twice as long as GB is just refer to the result that we did I think a couple of videos ago that the medians divided this triangle to just sit into six smaller triangles that all have equal area so another way to think about it is each of these three small triangles have you clear these are three of the total of six smaller triangles so these three all have equal area so let's think about this triangle right over here let's think about this triangle triangle a GB a GB this is triangle a GB right over there those are the same triangles now let's compare that to triangle let's compare it to triangle II AG right over here let's compare it to this triangle which is this triangle right over here on the original drawing now they both have the exact same height if you view eg is there as their base or not or they I guess they're they're shared base there's not they don't have the exact same base the smaller triangle has the base e if sorry the smaller orange triangle has GB as its base the larger blue triangle has E G as its base but they definitely both have the same height or altitude when you draw it this way so their height in both cases is this right over here now the other thing with that we do know the other thing that we do know is that this blue triangle e AG has twice the area of the orange triangle how do we know that because it's got two of these triangles in it so one way to think about it is if this orange triangle has area X so you let me call it a well I too used a so I'm gonna call it area X then each of these blue triangles have area X or you could say this entire blue region this entire blue region has area 2 X area 2 X so if you look at this blue triangle right over here we know that 1/2 times base times height is equal to area so we get 1/2 the base is eg 1/2 in the green color 1/2 EG x height x this yellow height is going to be equal to 2x is going to be equal to 2x I'm just I'm just applying the formula for area of a triangle 1/2 base times height is equal to area this is our area now let's do the same thing for this orange triangle 1/2 let me scroll over a little bit to the right we have 1/2 G B G B times the yellow height times height is going to be equal to X is going to be equal to X well we can substitute and this is equal to X we can we can place this entire expression right over here for X so let's do that we get 1/2 and you might already see where this is going but I won't I won't skip any steps here we get 1/2 times EG x EG times H times H is equal to 2 is equal to 2 times X but instead of X I'm going to write this here is equal to 2 times 2 times one half times G B times this length times the base of the smaller triangle times H times H and now we can just simplify this we have 2 times 1/2 is just going to be it's just going to be 1 you can divide both sides by H and we are left with 1/2 eg is equal to GB or we could write eg over 2 is equal to me to the same color since I've gone this far with the same colors so we can write we can write 1/2 eg eg is equal to GB is equal to G and we're done this is essentially saying that GB is half of eg GB is half of eg so for example if eg is 2 this is going to be 1 if eg is 4 this is going to be 2 so we've actually proven our result well actually let's go back to this is the result we wanted to prove to get to there we just to multiply both sides of this equation by 2 you multiply this the left-hand side by 2 you get eg you multiply the right-hand side by 2 you get gb so we've proven that eg is twice GB and you can apply the same logic to any of the medians to show that the centroid is exactly 2/3 along the way of the median