So what we have is we have a triangle
The only requirement is that the base angles here are acute.
And what we want to do is we want to somehow either draw a square,
so I'm just going to roughly sketch it out. In this case
we can see that, you know, we have a rectangle that is slightly wider than it is high
so
How can we get a square into this triangle, using just a compass and straightedge?
that lies on this base and then touches the two sides.
As mathematicians, before we start on actually constructing something, a good question to ask is:
Does this exist?
You know, how do we know we can get exactly a square, instead of just a rectangle?
So what we're going to do, is actually use this idea of rectangles. So, as mathematicians,
sometimes we are looking for squares, but we can't find a square,
so we relax the restriction and we start looking for rectangles and then we hope we can find a rectangle
where the height is equal to the base length, and then we know we have a square.
So, how can we get rectangles?
Well from the height of this triangle we can just draw any
parallel line to the base
and then drop the perpendiculars
and we now get a rectangle. We see that this blue rectangle is now a lot higher than it is long.
The green rectangle was a lot longer than it is high
so we hope that there's a square that lies somewhere in between this range.
(Brady: As you sort slide between the two?) - Right.
So let's actually draw it out. You know what that looks like, so I'm going to draw a graph.
Actually I'm using a ruler to measure stuff. So this isn't yet a purely straightedge and compass construction
So we're going to plot the height of the rectangles as they get lower and lower.
So for now I'm going to use my ruler to draw it out.
So it is off about
4 units
so over here is going to be about four units and
at the height of zero the height of the rectangle is zero,
so what we have is a straight line here
representing the height of the rectangle so
Now that we have the height of the rectangle, we're going to draw the width of the rectangle.
When the height is at the full height the width of the rectangle is - zero
So ever point down here and when the height is at zero, we have the full length of the base.
Which means we're going to have a straight line here.
And.. what happens at this point of intersection?
Well at this point of intersection, we know that the height of the rectangle is equal to the width of the rectangle
and so that gives us a square. (Brady: So we can do a square?)
- So, we can do a square and
when we had the use of ruler to transfer heights and lengths around we could find this is the height at which
the square will occur.
So if we have a proper construction, the hope is that we can draw a square here.
(Brady: Oh, yeah)
- So we'll see if we actually can match them. Another trick that mathematicians use is that if they can't find a square
they actually force out a square
So, how do I force out a square here, is that it's hard to force a square in there
But I can draw a square down below.
So using my compass and straightedge I can actually
draw a square and we're going to use this
to
Approximate a compass. So over here, we want to draw a square
It's going to come down to here
over here
We want to draw a square,
it's going to come down to - about here.
And so this gives us an approximate square.
And actually I do not need this ruler anymore, so I'm going to put it away and just use a straight edge.
So we're going to
continue this out
and
continue this out. And now what we have here, is we have a square, in this large triangle.
So it's not quite the square that we want in this small triangle up here, but at least it's a square in a large triangle.
What can we say about this small triangle and this large triangle?
Because this edge moves down parallel, these two triangles are actually similar
So there's a mapping that brings us from this small triangle,
to this large triangle. And we can ask: "What does the mapping do to this square?"
(Brady: Because you've got the square now?)
Yes! So think about what happens to the base of this square,
as we expand the small triangle to the big triangle, and as we contract it back.
This base is actually just going to be a straight line,
through to the vertex of the original triangle.
Likewise over here,
if we connect the vertex to this point, is how this point will move as we contract the triangle
So now the base of the large square
goes over to these two points. And what that means,
is that we now have the base of a smaller square in this triangle up here.
We just need to construct the
perpendiculars and
this will give us our square. And as I mentioned earlier we can compare it to this diagram on the right
And we can see that we are
reasonably close to what the calculator answer is.
Method one, where we constructed the two lines that intersect each other to find the height of the
square. And method two, where we forced a square to get into this triangle.
So now we're going to look at a third way -
How else can we relax the restriction of the square?
You know, instead of forcing all the vertices to line up perfectly on the edge of the triangle,
maybe we could say hey,
let's just have
the base of the square touching
one side of the triangle. Okay, we can also continue drawing more squares,
and see what they look like.
Okay, and so this figure of squares that are touching
the side of this triangle,
we hope we eventually will get to a square that does indeed touch this side of a triangle.
If we can find such a point we are done. And what we have here, is that we again have our similar triangles.
So we started out with this triangle,
and we moved the base over we got to this triangle.
So what then does that tell us about this point,
and
the vertex of this triangle, is that these three points will form a straight line.
So, we could
connect up these three points, and see where it intersects the other side of the triangle
and then from here, we know if we draw the perpendicular and we drop the parallel line that will give us a square.
(Brady: Huh. That's even.. that's almost simpler than the other one.)
That is simpler than the other one
And you know, both of this use the same idea of similarity.
[paper crunching noises]
Alright, so we are back on the first diagram and remember what we wanted to do is start off by constructing a square.
So,
I'm going to just quickly measure off
And we say, that,
the vertex of the triangle, and this corner of
our now relaxed square, we want it to
we want to extend this line and we can see that - hey, it does! -
go through this other point on a triangle. That's pretty amazing :D
(Brady: Look how happy you look. You're very happy with yourself.) [laughs]
So we have three different methods here and they all agree on this one point to give us the square,
we drop the perpendicular, we drop the parallel line, and we form our square in this triangle
[Preview of next video] Get wild
woohoo
So it looks like, for any triangle we will have those three segments intersecting in a point
Well that point turns out to be the center of mass, or our point, which also is known the centroid.