The only requirement is that the base angers here, acute.

And what we want to do this.

We wants to somehow other 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 as slightly wider than it is high.

So how can we get a square into destroy anger, using just a compass and strange that lies on this base and then touch us the two sides as mathematicians before we start on actually constructing something?

A good question.

Us is dust.

This exists, 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 off tangos.

So as mathematicians, sometimes we're 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 off this.

Try anger.

We can just draw any parallel line to base and then dropped the perpendicular us.

And we now get a rectangle.

We see that this blue rectangle is now a lot higher than it is long.

The green rectangle waas a lot longer than it is high.

So we hope that that's a square that lie somewhere in between this ridge and you sort of 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.

Leslie, I'm using ah, ruler to measure stuff.

So this isn't yet a purely trick each and campus construction.

So we're gonna 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's off about four units, so over here is going to be about four units and at a height of zero, the height of the rectangle zero.

So what?

We have this a straight line here representing the height off the rectangle.

So now that we have the height of the wrecked anger, we're going to draw the whiff off the rectangle.

When the height it's at the full height, the width of the rectangular is zero.

So have a point down here.

And when the height is that zero, we have the full length off the base, which means we're gonna 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 wrecked anger is requested a whiff off the rectangular, and so that gives us a square so we can do a square so we can do a square.

And when we had the use off this ruler to transfer height and length around, we could find this is the height at which the square walker.

So if we have a proper construction, the hope is that we can draw a square here, so we'll see if we actually can match that.

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 the 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 straight itch.

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 gonna 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 don't need this rule anymore, So I'm gonna put it away and just use a straight age.

So we're going to continued this out and continue this out.

And now what we have here is where a square in this last try anger.

So it's not quite the square that we want in this mall trying up here.

But this is a square in a large triangle.

What can we say about this More?

Try anger.

And this last try anger because this itch moved down.

Parallel this to trying us actually similar.

So there's a mapping that brings us from this more.

Try anger to this last rancor and we can ask what does the mapping due to this square?

Because you've got the square now?

Yes.

So think about what happens to the base off this square as we expand a small try anger to the Big Trang er and s to contract it back.

This base is actually just going to be a straight line.

True to the Vertex, off the originator Anger.

Likewise.

Over here, if we connect the Vertex to this point is how this point will move as we contract the trying.

So now in the base of the last square goes over to this two point And what that means is that we now have the base off a smaller square.

In this trying up here, we just need to construct the perpendicular us and this will give us all 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 complete answers Method one where we constructed the two lines, the intersect each other to find the height off the square.

Admit it to where we forced a square to get into this triangle.

So now we're going to look at 1/3 way.

How else can we relax the restriction off the square?

You know, it's that are forcing off the Vertex to line up perfectly on the age off the try anger.

Maybe we could say, Hey, let's just have the base off the square, touching one side off the triangle.

Okay?

We can also continue drawing more squares and see what they look like.

Okay?

And so this figure off squares, they are touching the sight of this.

Try anger.

We hope we eventually will get to a square that Dustin did 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 how similar triangles.

So we started out with this triangle and removed the base over.

We got to destroy anger.

So what, then does that tell us about this?

Points and the vertex off this triangle is that this tree points we'll form a straight line so we could connect up this tree points and see where it intersects the other side off the try anger.

And then from here, we know if we drop the perpendicular and we dropped the parallel line, that will give us a square.

Ah, that's even.

That's almost simple.

Of the other one that isn't simply it And the other one and you know both off this use the same idea off similarity.

All right, so we're back on the first diagram.

And remember what we wanted to do.

It ISS start off by constructing a square.

So I'm going to just quickly measure off.

And we said that the vortex off the try anger and this corner off I were.

Now relax, Square.

We wanted to You want to extend this line, and we can see that.

Hey, it does go through this at a point on a triangle.

That's pretty amazing.

Okay, you're very so we have traded from Method CIA and they all agree on this.

One point to give us the square we dropped the perpendicular would drop the parallel line.

Every farm I'll square in this triangle wild.

So it looks like any triangle we will have those three segments intersecting in the point.

Well, that point turns out to be the center of mass or our point, which also is known the central.