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## High school geometry

### Unit 8: Lesson 12

Constructing circumcircles & 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.

## Want to join the conversation?

- The question asks to construct a circle inscribing the triangle.

So doesn't that mean the triangle should be inside the circle ?

The opposite goes for the next VDO : Constructing circumscribing circle(22 votes)- It is an "inscribing" circle because the circle is inscribed inside the triangle. Agreed though, "Construct a circle inscribed in the triangle" would be more clear.(6 votes)

- The use of the 4 headed arrow is sub-optimal. Much better would be a hollow circle with short horizontal and vertical lines (about 1/2 radius) pointing inward so you could see exactly where you are placing the dot. Further: The drawing should be closer to the 'ideal' line and point.

This is important as multi-stage constructions have accumulated errors. E.g. in the video on an inscribed circle the speaker demos how to do it, and despite exercising reasonable care, his circle doesn't quite touch one side. This both introduces the idea that sloppy workmanship is tolerable, and introduces doubt as to the validity of the construction.

Also: There is no explanation as to why the construction is valid. Why is the intersection of two bisectors the centre of the circle? By what magic is this the right place for the third side.

The proof is not difficult, although it has many steps:

Bisect the angle.

Pick a point on the bisector.

From that point construct perpendiculars through that point to each of the two sides of the angle.

Show that the two triangles formed are congruent.

Since the point is arbitrary, it means that any point on the bisector is equidistant from both sides of the triangle.

Repeat for another angle.

Repeat the construction from the intersection to all sides. One of the perpendiculars will be a side of two different triangles. Equality is transitive so if A=B and B=C then A=C so all three lengths are equal.(6 votes)- I agree that this does introduce the idea that "not quite", or "almost" is tolerable, but sometimes it is. The point here is to teach you how to do it, not to be a perfectionist. Khan academy usually teaches one method, the one they think is best. There are many different methods and proofs you can find online. If you can do a proof yourself, why does someone need to show you? Good question Sherwood.(0 votes)

- How does Sal know to take the two circles and place them like that? How does he know it will bisect that angle?(2 votes)
- Sal kind of skips the normal first step of starting at a vertex and drawing a circle, then drawing two congruent circles whose intersection would be equidistant from the two points. Sal avoids the first step by drawing two congruent circles whose radii are equidistant from the vertex. Does this help, or you need more explanation?(1 vote)

- Why doesn't Sal use the perpendicular bisector method, by putting a compass on either vertex, to create the bisector?(2 votes)
- The point is to bisect the angle, no the line. Occasionally it does both at once, but not always. Although he is bisecting the line, that is not the goal.(1 vote)

- What can j be doing wrong? I do all the steps (double checked ) and yet my circle is never the right size.(2 votes)
- Can you construct a circle that is inscribed in a square?(1 vote)
- Yes. Take a square with side length x. Draw the two diagonals. Put the compass at the intersection of the two diagonals and draw a circle with radius x/2. This will be the inscribed circle.(3 votes)

- How do you construct a circle in a triangle using a compass?(1 vote)
- as far as I know, we can't do that. Just draw the best circle you can.(1 vote)

- When the question asks to construct a circle inscribing the triangle, does it want the largest possible circle, or a circle of any size?(1 vote)
- I haven't watched any videos about the intersection of triangle/s angle bisectors in High School Geometry. Where can I find the video proof?(1 vote)
- how do we prove it using congruent triangles?(0 votes)

## 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.