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Geometric constructions: triangle-inscribing circle

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

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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.