三角形重心與中線 (2D 英語證明)

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三角形重心與中線 (2D 英語證明) : 重心位於每條中線的2/3。

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In the video on triangle medians and centroids
I did essentially the proof that the centroid is 2/3 along the way of the median.
I did it using a two-dimensional triangle in three dimensions, and I mentioned that I thought at least it made the math a little simpler
Ah, but someone mentioned that they'd be interested in seeing the two dimensional version of the proof,
so, so why not do that. So let's just draw an arbitrary triangle,
and I'll make one, I'll make one side of the triangle right along the x- axis
just to make, I think it'll make the math a little bit easier,
maybe there's easier ways of doing it.
So that's one side of the triangle, that's the other side of the triangle.
And let's put, let's put kind of the height of the triangle, let's put it on the y-axis.
So this triangle right here, this is y-axis, that is the y-axis, and then
this right here. This right here,is our x-axis. Now, we said that this is an arbitrary triangle.
So let's make, let's just make this distance equal to A, so this point right here, this vertex would be the point (A,0), so that distance is A, let's make
this distance, let's make this distance right here be equal to, I don't know let's make it equal to B
so this is the point (-B,0)
and then let's make this distance, this is the point Y=Z, so this is the point (0,Z) So this is an arbitrary triangle, any triangle can be represented this way
Now let's think about it's medians, let's think about medians and the centroid And I'm only gonna do for two of the medians, because we know
that the third median will also intersect at the same centroid
So let's just find, so we have a median on this, we have a midpoint,
I should say, on this side of the triangle
And it's coordinates are just gonna be
midpoint of those two points or,
zero plus A over 2, which would just be A over 2
And then C plus 0 over 2, which would just be C over 2
And let's do the same thing over here
The midpoint of this side, right over here, is going to be,
negative B plus 0 over 2
So it's negative B over 2
And then, 0 plus C over 2
So it's just going to be, it's just going to be C over 2
Now that we know all the coordinates for the vertices
and the midpoints,
at least for this two sides, we can find the equations
for the lines that the medians are part of
So we could find the equation for this line, we could find
the equation in this line in purple, then we could find the equation
for the other lines that connect this 2 dots, find the intersection,
and we'll essentially have the coordinates of our centroid,
of the intersection of the medians
And we could do it for this one too,
but that would be redundant because
it's going to intersect in the same point
So what would be, what would be the equation
for this line right over here?
Well our slope, our slope is going to be changed in Y,
so C over 2 minus 0,
so that's just C over 2, over changing in X
So, A over 2 minus, minus B, minus B
So it's a minus -B I should say
So, it's plus B, could write a plus B over here,
but just to make the math simple,
I'll write plus 2B, plus 2B over 2, right?
I just subtracted a -B, which is the same thing as adding a B,
and I just, and adding B is the same thing as adding a 2B over 2
And I did that so these have the same,
these have the same denominator
I could add them, or I could just multiply the numerator
and the denominator both by 2,
and I'll get the slope is equal to C over A plus 2B
So that's the slope of this line, right over here
And then, we know some points here, we could just use
the point slope formula for the equation of a line
We know the point -B comma 0 is on the line,
so we know that the equation of this line in purple,
let me state in the purple is going to be, Y minus 0
is equal to, is equal to X minus negative B,
or I could say X plus B times our slope,
times C over A plus 2B, A plus 2B
And obviously, this would just simplify that Y
is equal to all of these business over here
So this is essentially the equation of this median right here,
if the line just kept going, or this line
The median is a line segment that is part of this line
Now let's do the same thing for, let us do the same thing for
this line right over here
This median right over here
So once again, its slope is changed over,
change in Y over change in X
So the change in line C over 2 minus 0
So the slope here is equal to C over 2 minus 0,
which is just C over 2,
over -B over 2, -B over 2 minus A
Or I could say, minus 2A over 2,
that's the same thing as minus A
And do the same thing, multiply the numerator
and the denominator by 2
We get C over, C over -B minus, -B minus 2A
So the equation of this line right here, the other median,
the equation of this other median right over here is,
we have this point over here so, we could use the point slope again
We get Y minus 0, Y minus 0, is equal to X minus A,
is equal to X minus A times the slope
Time C over -B minus, -B minus 2A
So we have the 2 equations for these lines, we can now use
that information to find their intersection
which is going to be, the centroid
Well to do that, we have both of these in terms of Y right?
Y minus 0 is the same thing as Y, Y minus 0 is the same thing as Y
So we could just set this as being equal to that
So let's do that
We get, X plus B times C over A plus 2B is equal to, is equal to
X minus A times C over -B minus 2A
Well both sides are divisible by C,
we could assume that their C is non-zero,
so this triangle actually exist
It actually exist in 2 dimension
So we can divide both sides by C and we get that
Now let's see, now we can cross multiply,
we can multiply X plus B times this quantity right here
and that's going to be equal to A plus 2B
times this quantity over here
And so we can just distribute it, so it's going to be
X times all of this business,
so it's going to be X times -B minus 2A
plus B times all of these
So that gonna be minus B squared, minus 2AB
So i just multiplied that times that,
is going to be equals to that times that
So it's gonna be X times all of this business
X times A plus 2B, minus A times that
So minus, distribute the A, minus, A squared minus to AB,
see if we can simplify this
So it's, we have a negative 2AB on both sides,
so let's just subtract that out
And then if we, let's, so we cancel things out,
let's subtract this from both sides of the equation,
so we'll have a minus, minus X times A minus 2B do,
and I'll subtract that from here,
but I'll write it a little different,
it'll simplify things
So this will become, this will become a X times, I'll distribute
the negative inside the A plus 2B
So negative A minus 2B
And let's add a B squared to both sides
So plus B squared, plus B squared
I want to collect all the X terms on one side
and all the the constants on the other
And so this becomes, this becomes
On the left hand side the coefficient on X,
I have negative B minus 2B is,
negative 3B, negative 2A minus A is, negative 3A times X,
is equal to, these guys cancel out,
these guys cancel out, is equal to
B squared minus A squared
Is equal to B squared minus A squared
Let's see if we can fact,
this already looks a little suspicious in a good way
Something that we should be able to solve
This can be factored, we can factor our negative 3
or we can actually factor out a 3,
we'll get 3 times B minus A, 3 times, actually,
let's factor a negative 3
So negative 3, negative 3 times B plus A times X is equal to,
now this is the same thing as B plus A, times B minus A!
So we can divide both sides by B plus A, we have negative 3 times X
is equal to B minus A, we'll just divide both side by negative 3
We get X is equal to B minus A over negative 3,
which is the same thing asA minus B, A minus B over 3
So we have the X coordinate,
we have the X coordinate so far for our centroid
It is A minus B over 3
It is A minus B over 3
And you kinda see hints of the solution over here
A minus B is its entire distance
It's one third of this entire distance, although,
I don't want to jump the gun too much because
it complicates it just thinking about, just the X coordinated
Let's figure out the Y coordinate
To do that, we could just substitute back
into one these equations up here
This was the, this was the equation of this median,
the gray median that we had done
So let's substitute back in, we have Y is equal, Y, I did over here
Y is equal to X minus A times C
X is this thing over here
So it's A minus B over 3 minus A
So I'm gonna subtract this, instead of just writing
a minus A over here, I'm gonna write negative 3 over 3
So minus 3A over 3
So that's just minus A right over here
I just multiplied it and divided it by 3
It's gonna be all of that times C over, over negative B minus 2A
So this is going to be equal to C
In the numerator up here, we have an A minus 3A,
so this equal to negative B, negative B
We have a negative B there
And then we have aA minus 3A is minus 2A
So this becomes negative B minus 2A times,
we could factor out the one third,
times C over 3, so that's what the numerator is
All of that over negative B minus 2A
Well that's going to cancel with that
And so, our Y coordinate is going to be equal to C over 3
So our Y coordinate here is C over 3
Now could just use the distance formula now
We know all the coordinates and,
and you feel free to do it if you like
But there's a slight simplification,
or at least in my mind, a slight simplification
We know, we know that the height, we know that the Y coordinate,
we know that the Y coordinate of the centroid is C over 3
So this right here, this point over here is C over 3
So we could actually use, maybe an easier argument, cause remember,
our whole point was to show, our whole point was to show
that this point is two thirds along the median away from the vertex
So let me draw this vertex now here
Our whole point of this video as we said,
this is the midpoint of this side over here,
is to show that this, this right here is
two thirds along the way
Or that this length is twice this length
Or that this length, this length is two thirds
of the entire length of the median
So how can we do that?
Well that simplest way I can do that, I mean you actually,
you could actually just use the coordinates
and use the the distance formula
But a simpler way, just the way that we've arranged this,
is just to use an argument
of similar triangles
Cause this right here, this right here is,
we know that this entire height right here in blue is C
We know this entire height over here in blue is C
We know that, we know that this height
right over here is C over 3 or,
you could do that as one third times C
And so, we know that this height,
we know that this height over here,
we know that, that distance is two thirds
Is two thirds, is two thirds C
So we can use a similar triangle argument
Let's say, let's say that this entire thing right here
is the length of the median
And let's just call that, I dunno, let's call that L
And let's say we wanna figure out how far along the median away
from the vertex that is
And let's just call that, I dunno,
let me use another, another,
let me just call that distance right over,
let me do, use a different color
Let's call, let's call this distance right over here,
you see, I've already used A, B's, and C's,
let's call that D
So I, if we can show that the ratio
of D to L is two thirds
So this is two thirds of the distance then, we're done!
And use that by similar triangles
we have two similar triangles
We have this, lemme use a new color
We have this triangle, which is kind of embedded inside
of this larger triangle right over here
The both share the same vertex, this angle over here
They both have right angles over here
It's as they have two of the same angles,
the third angle has to be the same
So they're definitely similar triangles
so we could just set up a ratio here
Two thirds C, which is this length right over here
That's the, that's just kind of the,
the vertical side of the smaller triangles
We could write two thirds C over this entire length,
over the length of the larger that's D
This corresponding side of the larger similar triangle
over C is equal to, is equal to
The hypotenuse of the smaller similar triangle D
over the hypotenuse of the longer,
of the longer similar triangle, or the bigger similar triangle
D over L
Well this is clearly, just divide the numerator and denominator by C
this clearly becomes two thirds
So the ratio from, of this, of this, of this length
to this larger length, is two thirds
Or this is two thirds a long way,
a long the median away from the vertex
So there you go
That's the two dimensional proof that the centroid
is two thirds along the way
of any median from the vertex, or one third along the way,
along the median
Or one third of the distance of the median
from the opposite midpoint
Anyway, hopefully you enjoyed that

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