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# Geometric constructions: circle-inscribed equilateral triangle

CCSS Math: HSG.C.A.3, HSG.CO.D.13

## Video transcript

Construct an
equilateral triangle inscribed inside the circle. So let me construct
a circle that has the exact same dimensions
as our original circle. Looks pretty good. And now, let me
move this center, so it sits on our
original circle. So they now sit on each other. Or their centers now
sit on each other. So I can make it, and
that looks pretty good. And now, let's think
about something. If I were to draw this
segment right over here, this, of course, has the
length of the radius. Now, let's do another one. And that's either
of their radii, because they have
the same length. Now, let's just center
this at our new circle and take it out here. Now, this is equal to the
radius of the new circle, which is the same as the
radius of the old circle. It's going to be the
same as this length here. So these two segments
have the same length. Now, if I were to connect
that point to that point, this is a radius of
our original circle. And so it's going to have
the same length as these two. So this right over
here, I have constructed an equilateral triangle. Now, why is this at all useful? Well, we know that the angles
in an equilateral triangle are 60 degrees. So we know that this angle
right over here is 60 degrees. Now, why is this being
60 degrees interesting? Well, imagine if we constructed
another triangle out here, just symmetrically, but kind of
flipped down just like this. Well, the same argument,
this angle right over here between
these two edges, this is also going
to be 60 degrees. So this entire interior
angle, if we add those two up, are going to be 120 degrees. Now, why is that interesting? Well, if this interior
angle is 120 degrees, then that means that this arc
right over here is 120 degrees. Or it's a third of the
way around the triangle. This right over here is a third
of the way around the triangle. Since that's a third of the
way around the triangle, if I were to connect these
two dots, that is going to be, this right over
here is going to be a side of our
equilateral triangle. This right over here,
it's secant to an arc that is 1/3 of
the entire circle. And now, I can keep doing this. Let's move-- I'll reuse these--
let's move our circle around. And so now, I'm going to move
my circle along the circle. And once again, I just want
to intersect these two points. And so now, let's see, I
could take one of these, take it there, take it
there, same exact argument. This angle that I
haven't fully drawn, or this arc you could
say, is 120 degrees. So this is going to be one side
of our equilateral triangle. It's secant to an
arc of 120 degrees. So let's move this around again. Actually, we don't even have
to move this around anymore. We could just connect
those last two dots. So we could just
connect this one. Actually, I just want to, let's
connect that one to that one. And just like that,
and we're done. We have constructed our
equilateral triangle.