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## Geometry (all content)

### Unit 16: Lesson 3

Constructing circumcircles and incircles# Geometric constructions: triangle-inscribing circle

CCSS.Math:

Sal constructs a circle that inscribes a given triangle using compass and straightedge. Created by Sal Khan.

## Video transcript

Construct a circle
inscribing the triangle. So this would be a circle that's
inside this triangle, where each of the sides
of the triangle are tangents to the circle. And probably the
easiest way to think about it is the center
of that circle is going to be at the incenter
of the triangle. Now what is the incenter
of the triangle? The incenter of the
triangle is the intersection of the angle bisectors. So if I were to make a
line that perfectly splits an angle in two-- so I'm
eyeballing it right over here-- this would be an angle bisector. But to be a little bit more
precise about angle bisectors, I could actually use a compass. So let me make this
a little bit smaller. And what I can do
is I could put this, the center of this circle, on
one of the sides of this angle right over here. Now let me get another circle. And I want to make
it the same size. So let me center it there. I want to make it
the exact same size. And now let me put
it on the other one, on the other side of this angle. I'll put it right over here. And I want to put it so that
the center of the circle is on the other
side of the angle, and that the circle
itself, or the vertex, sits on the circle itself. And what this does
is I can now look at the intersection of
this point, the vertex, and this point, and that's
going to be the angle bisector. So let me go-- I'm going
to go through there and I'm going to
go through there. Now let me move these
circles over to here, so I can take the angle
bisector of this side as well. So I can put this one over here. And I could put this
one-- let's see, I want to be on the
side of the angle. And I want the circle to go
right through the vertex. Now let me add another
straight edge here. So I want to go
through this point and I want to
bisect the angle, go right through the other point
of intersection of these two circles. Now let me get rid of
one of these two circles. I don't need that anymore. And let me use this one
to actually construct the circle inscribing
the triangle. So I'm going to put it at
the center right over there. Actually, this
one's already pretty close in terms of dimensions. And with this tool, you don't
have to be 100% precise. It has some margin for error. And so let's just go with this. This actually
should be touching. But this has some
margin for error. Let's see if this
was good enough. It was.