But what I think about now is the if we're given some triangle,
and here we have triangle ABC
What are the medians of the triangle
and how do they relate to each other
and do they have some interesting properties,
and you might guess that they do
So the median is, if we start at one of the vertices
so let's start at this one right over here
And then we bisect the opposite sides
So this right here would be a median
And so it started there, and it bisects this side right over here
So the length from B to let's call this D
is equal to the length from D to C
Now, let's do that with every side
I can draw a median over here, like that
So, let's call this point E
So the length from A to E
is going to be equal to the length from E to C
Although it's kind of a little lopsided,
but that gets it pretty close
And then we're gonna draw another median
And I'm not gonna prove it in this video,
but all the medians, and this is another neat thing, that, you know,
when you have three lines that will always intersect at one point,
that all the medians do intersect at one point
They all are concurrent
They all hit one common point in the center
So, let's draw it that way
I'm not gonna prove it in this video, so this length,
let's call this point right over here F
So this length right over here,
this length right over there is equal to that length over there
And the point at which these medians intersect
is called the centroid, the centroid
And when you start studying physics,
if you're actually and this was a uniform triangle
and you were to throw it,
it would rotate around the centroid right over here
But we'll study it geometrically for now
So let's call this ten centroid well, we've already gone up to F
So let's call this centroid G right over there
Now, what I wanna that by itself is neat that you have a centroid,
that if you were to throw it, if it was a uniform mass,
it would rotate around the centroid
But what's even neater about this,
is that we can see that we've divided this triangle
into six smaller triangles
What's really cool,
even though these aren't congruent triangles necessarily,
but they do have the same area
And that's what we're going to prove in this video
That these six triangles all have the same area
Now, to start off, I'm just going to look at two
I'm gonna look at different pairs of triangles
So let's look at let's look at these two triangles right over here
Let's look at those two triangles over here
And to show that those two have the same area,
we're just going to invoke a very simple principle
So, imagine rotating, just those two triangles over
So just rotating, just those two triangles over
It would look something like this
Let me try my best to draw it
Where this would be point G I'll even try to color code it the same
that is point G
That is that side right over there
This is point C
This is point B, that is point B
And then this right over here
would be the second part of that median right over there
That over there would be point D
Now, we know and I didn't draw it I should, at nicely over here
We know that this length is equal to this length right over here
And these two triangles, if we're starting to think about the area,
they have the same base
And we know area
Area is equal to ? base * height
So, they definitely have the same base
What about their heights?
Well, they also have the same heights
Both of their heights, both of their height is exactly that tall
They both have the same height
So both of them have the same base
They both have the same height
On an ob on an obtuse triangle right here, the altitude sits outside of it
So that might be a little counterintuitive
So if you have an obtuse triangle like that
I say obtuse cause this is more than 90 degrees
Your altitude, your height will actually
sit outside of the triangle, but that's okay
Both of these triangles have the same base, and the same height,
so they must have the same area
So if this one right here has area x,
this one right here will have area x as well
And you can use the exact same logic to say, well, look
This guy and this guy, they have the same base,
and they both have the same height
So if this one right over here is of area y,
then this one over here is also going to be of area y
They will have the same area
And then finally,
we could do the same thing for these two characters up here
They both have the same base
This was BF is equal to FA, and they both have the same height
We drop an altitude like that
And so if we call this area, if we call this area right over here Z,
you could call that area Z, as well
So, so far, we've shown that we can divide this
into three pairs of triangles that have the same area
But we wanna now show that they all have the same area
And to do that, we can invoke this same principle,
but we'll do it with different sets of triangles
So now, let's look at triangle let's look at triangle BAE
Look at triangle BAE, triangle BAE
So the area of triangle BAE,
area of BAE, is going to be equal to z + z + y
z + z + y
And let's look at the area of triangle BEC, BEC right over there
That's going to be,
this triangle right over here, is going to be x + x + y
So the area of BEC is going to be x + x + y, but the same principle
They both have the same base, and they both have the same height
We could drop an altitude like this
This one is obtuse, so that the altitude sits outside of it,
but they have the exact same height
So these two areas need to be equal to each other
So you have Z, well, let me just add that up
Now you have 2z + y is going to be equal to,
is going to be equal to 2x + y
2x + y
Subtract y from both sides
You get 2z is equal to 2x
Divide both sides by 2
You get z is equal to x
And so we could say, we could write an x here, and an x there
So we know that all of these will have the same area,
but we still have to worry about these y's here
And to do that,
we just have to kind of rotate the way that we look at it
And now look at triangle ADC
Let me do that in a different color
Triangle ADC, which I'm highlighting right over here
Triangle ADC, whose area, so the area of ADC is going to be 2y + x,
is equal to 2y + x
And then look at triangle let's see, what color have I not used yet
Let me use this green
Triangle ADB, triangle ADB
Triangle ADB is going to be equal to,
well, you could say it's 2z + x,
but we know that the z's are equal to x
It's really just x + x + x
It's really equal to ADB is equal to 3x
And we have the same idea here
ADB has this base, which is the same as ADC's base,
and they both have the same height
We can draw up an altitude like this,
we can draw up an altitude like that
They have the same height
We're just invoking this principle over and over and over again
So these things have to equal each other
So we get 2y + x is equal to 3x, is equal to 3x
Subtract x from both sides
You get 2y is equal to 2x, or, or y is equal to x
So that's a really neat result
You go from each of the vertices of a triangle to the opposite side,
and you bisect that opposite side, and you do that three times,
you have three medians
These lines are called medians, where they intersect as a centroid
You know what's really cool is,
it divides the triangle into six smaller triangles of equal area