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## Geometry (all content)

# Centroid & median proof

Showing that the centroid divides each median into segments with a 2:1 ratio (or that the centroid is 2/3 along the median). Created by Sal Khan.

## Want to join the conversation?

- Is this really a textbook "proof"? It seems that my textbook doesn't want proofs to take the form of a set of equations. I see that this does "prove" the point, but my book wants a list of theorems and postulates and such. HELP!!!(4 votes)
- I know. In a real test, sadly, you will probably have to list all those theorems and postulates, even completely obvious ones, (such as the fact that there is a postulate for x=x)! Sal, is doing this proof to make it easier to understand this topic.(16 votes)

- Doesn't he show this in a previous video as well?(9 votes)
- Yes, Triangle Medians and Centroid. He solve it with a three dimensional plot (x,y,z) axes. The video following that showed how to solve it in 2D which is more difficult.(5 votes)

- at0:01sal says " i have drawn an Arbitrary triangle" what's an Arbitrary Triangle?(2 votes)
- an Arbitrary triangle is a triangle that has no definite side lengths, no definite angles, and the vertices have no definite position. In other words, it is equally likely to be ANY POSSIBLE TRIANGLE.(6 votes)

- Why is the
*centroid*known as the**center of gravity**? Why isn't it the circumcenter, incenter, or any other point of concurrency? Why the centroid?(2 votes)- Because the centroid is the physical center of gravity. If you had a paper triangle, you could balance it on a pencil if you put the pencil under the centroid.(2 votes)

- Is there a relationship between the Circumcenter, the Inradius and the Centroid of a triangle? Would these three points all be the same in an Equilateral Triangle?(1 vote)
- Yes, all three points would be the same in an equilateral triangle. If the triangle was not equilateral, then the points would fall on the same line, known as the Euler line.(3 votes)

- How do we know for sure that the area for all 6 triangles inside the bigger triangle are equal?(2 votes)
- At3:59, how do we know that the two blue triangles together have twice the area of the orange triangle? We don't know the relationship between them other than the height, but the height doesn't even matter when you split of the blue triangles.(2 votes)
- In this video, http://www.khanacademy.org/math/geometry/triangle-properties/medians_centroids/v/medians-divide-into-smaller-triangles-of-equal-area, Sal proves that the 6 smaller triangles formed by the 3 medians all have equal area.(1 vote)

- what would be the reasons for each step(1 vote)
- Well most of the work in this proof was in showing the areas of the sub-divided triangles were all equal. That's done in this earlier video: https://www.khanacademy.org/math/geometry/triangle-properties/medians_centroids/v/medians-divide-into-smaller-triangles-of-equal-area

In this video the steps are just "formula for the area of a triangle" and then some basic algebra.(2 votes)

- so what's the difference between a median and a perpendicular bisector?(1 vote)
- Median - A line segment that joins the vertice of a triangle to the midpoint of opposite side.

Angle bisector - A line segment that divides an angle of a triangle into two equal angles.

Perpendicular bisector - A line segment that makes an angle of 90 deg (right angle) with the side of a triangle.

The common point where the medians intersect is the centroid.

The common point where the angle bisectors intersect is the incenter.

The common point where the perpendicular bisectors intersects is the circumcenter.(1 vote)

- But why is triangle AGE is twice the size of triangle ABG? Is there a way to prove it, and not speculate?(1 vote)

## Video transcript

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.