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# Geometric constructions: circle-inscribed regular hexagon

Video transcript

Construct a regular hexagon
inscribed inside the circle. So what I'm going
to do, first, I'm going to draw a
diameter of the circle. And actually, I'm going
to go beyond the diameter of the circle. I'm just going to
draw a line that goes through the center of the
circle and just keeps on going. And I'm going to make it flat,
so it goes directly through, so this one right over
here, goes directly through the center. And now, what I'm
going to do is I'm going to construct
a circle that's the exact same dimensions of
this circle that's already been drawn. So let me put this
one right over here. And let me make it
the same dimensions. And now, what I'm going to do
is I'm going to move it over, so that this new
circle intersects the center of the old circle. And these circles
are the same size. So notice, this center
intersects the old circle. And the new circle
itself intersects the center of the old circle. Now, the reason why
this is interesting, we already know that this
distance, the distance between these two centers,
this is equal to a radius. We also know-- we have a
straight edge-- we also know that this distance right
over here is equal to a radius. It's equal to the radius of
our new circle right over here. We also know that this
distance right over here is equal to a radius
of our old circle. And that they both
have the same radius. So this is a radius
between these two points. Between these two
points is a radius. And then, in between these
two points is a radius. So now, I have constructed
an equilateral triangle. And essentially, you have
to just do this six times. And then I'm going to
have a hexagon inscribed inside the circle. Let me do it again. So we'll go from here to here. This is a radius of
my new circle, which is the same as the
radius of the old circle. And I could go
from here to here. That's the radius
of my old circle. So I have another
equilateral triangle. Radius, radius, radius. Another equilateral triangle. I've just got to do this, I
have to do this four more times. So let me go to my original. Let's see, let me make
sure I can-- well, it's actually going to be
hard for me to-- let me just add another circle here,
to do it on the other side. So if I put the
center of it right-- I want to move it a
little bit, right over-- I want to make it the same size. So oh, that's close enough. And let's see, that
looks pretty close. That's the same size. Now, let me move it over here. Now, I want this center to be
on the circle, right like that. And now, I'm ready to draw some
more equilateral triangles. And really, I don't have to
even draw the inside of it. Now, I see my six
vertices for my hexagon, here, here, here,
here, here, here. And I think you're satisfied
now that you could break this up into six equilateral triangles. So let's do that. So this would be the base of one
of those equilateral triangles. And actually, let me move these. Let me move this
one out of the way. I can move this one
right over here. Because I really just care
about the hexagon itself. And I can move this
right over here. But we know that these are
all the lengths of the radius anyway. Actually, I'm not even having
to change the length there. Then, I have to just connect
one more right down here. So let me add another straight
edge, connect those two points. And I would have done it. I would have constructed my
regular hexagon inscribed in the circle.