Centroid & median proof

I've drawn an arbitrary triangle right over here, and I've also drawn its 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 EB. What I want to do is I want to prove that EG is equal to 2 times GB. 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 EB. 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 median, 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 ABE right over here. And I'm going to draw this median as essentially the base. So let me draw it that way. I'm going to try to color code it similarly. So we draw it 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 have 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. 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 there is that marking, these two markings, these two markings are on this side right over there. And the whole way that I'm going to 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 divide this triangle into six smaller triangles that all have equal area. So another way to think about it is each of these three small triangles have equal area. 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 AGB. This is triangle AGB right over there. Those are the same triangles. And let's compare that to triangle EAG 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 as their base-- or I the guess their shared base, they don't have the exact same base. The smaller triangle has the base E-- sorry, the smaller orange triangle has GB as it's base. The larger blue triangle has EG 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 that we do know is that this blue triangle EAG 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, actually let me call it a-- well I already used a, so I'm going to call it area x-- then each of these blue triangles have area x. Or you could say this entire blue region has area 2x. 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-- I'll do that in the green color-- 1/2 EG times height times this yellow height is going to be equal to 2x. 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 GB times the yellow height is going to be equal to x. Well we can substitute it. If this is equal to x, 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 skip any steps here-- we get 1/2 times EG times h is equal to 2 times x. But instead of x, I'm going to write this here. Is equal to 2 times 1/2 times GB times this length-- times the base of the smaller triangle times h. And now we can just simplify this. We have 2 times 1/2 is 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-- let me do it in the same color since I've gone this far with the same colors. So we can write 1/2 EG is equal to GB. And we're done. This is essentially saying that 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 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.