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## Angle bisectors

# Incenter and incircles of a triangle

## Video transcript

I have triangle ABC here. And in the last
video, we started to explore some of the
properties of points that are on angle bisectors. And now, what I want
to do in this video is just see what happens when
we apply some of those ideas to triangles or the
angles in triangles. So let's bisect this angle
right over here-- angle BAC. And let me draw
an angle bisector. So the angle bisector
might look something-- I want to make sure I get
that angle right in two. Pretty close. So that looks pretty close. So that's the angle bisector. Let me call this point
right over here-- I don't know-- I could
call this point D. And then, let me draw
another angle bisector, the one that bisects angle ABC. So let me just draw this one. It might look something
like that right over there. And I could maybe
call this point E. So AD bisects angle BAC,
and BE bisects angle ABC. So the fact that
this green line-- AD bisects this angle
right over here-- that tells us that
this angle must be equal to that angle
right over there. They must have
the same measures. And the fact that this bisects
this angle-- angle ABC-- tells us that the measure of
this angle-- angle ABE-- must be equal to the
measure of angle EBC. Now, we see clearly that
they have intersected at a point inside of the
triangle right over there. So let's call that
point I just for fun. I'm skipping a few
letters, but it's a useful letter
based on what we are going to call this
in very short order. And there's some
interesting things that we know about I. I sits on
both of these angle bisectors. And we saw in the previous
video that any point that sits on an angle bisector
is equidistant from the two sides of that angle. So for example, I sits on AD. So it's going to be
equidistant from the two sides of angle BAC. So this is one side
right over here. This is one side
right over there. And then this is the other
side right over there. So because I sits on AD, we
know that these two distances are going to be the
same, assuming that this is the shortest distance
between I and the sides. And then, we've also shown
in that previous video that, when we talk about
the distance between a point and a line, we're talking about
the shortest distance, which is the distance you get if
you drop a perpendicular. So that's why I drew
the perpendiculars right over there. And let's label these. This could be
point F. This could be point G right over here. So because I sits on AD,
sits on this angle bisector, we know that IF is
going to be equal to IG. Fair enough. Now, I also sits on
this angle bisector. It also sits on BE, which says
that it must be equidistant. The distance to AB must be the
same as I's distance to BC. I's distance to
AB we already just said is this right over here. It's IG. But we also know
that that distance must be the same as the
distance between I and BC. So if I drop another
perpendicular right over here. And let's say I call
this point-- let's see, I haven't used H
right over here-- this distance must be
the same as this distance because I sits on this bisector. So IG must be equal to IH. But IF is also equal to IG. So we can also say that IF--
I mean, if IF is equal to IG is equal to IH, we also
know that IF is equal to IH. Pretty much common sense. If this is equal to that,
that is equal to that, then these two have to
be equal to each other. But if I is equidistant
from two sides of an angle-- this is the second
part of what we proved in the previous video--
if you have a point that is equidistant from
two sides of an angle, then that point must sit on the
angle bisector for that angle. So this right here
tells us that I must be on angle
bisector of angle ACB. Because it's equidistant to
those two sides of angle ACB. And what we have just
shown is that there's a unique point inside
the triangle that sits on all three
angle bisectors. It's not always obvious that if
you took three lines-- in fact, normally, if you
took three lines, they're not going to
intersect in one point. Two lines, a very
reasonable thing to do. But three lines, not always
going to intersect in one point. But once again-- like we
saw with the circumcenter where we took the perpendicular
bisectors of the side-- that was pretty neat that they
intersected in one point. Now, it's also cool
that we're showing that the angle bisectors all
intersect in one unique point. I is on the angle bisector of
ACB, so the bisector of ACB will look something like this. And this angle
right over here is going to be congruent to
this angle right over there. So we've just shown that
if you take the three angle bisectors of a triangle, it
will intersect in a unique point right over there that
sits on all three of them. So it seems worthwhile that
we should call this something special. And we do. And that's why I
called it I. We call I the incenter of triangle ABC. And you're going
to see in a second why it's called the incenter. When we talked about
the circumcenter, that was the center
of a circle that could be circumscribed
about the triangle. I-- we'll see in
about five seconds-- is the center of
a circle that can be put inside the
triangle that's tangent to the three sides. And how do we construct that? Well, we've just
established that I is equidistant to
each of the sides-- that this length is
equal to that length is equal to that length. So what happens if you
set up a circle with I as a center that has a radius
equal to the distance between I and any one of the
sides, which is equal, that has a radius
equal to IF, IG, or IH? Well, then, you're going
to have a circle that looks something like this. Let me draw it a little
bit better than that. Well, you can imagine. This is my best attempt
to draw a circle. This circle right here
that has the radius equal to the distance between
I and any of the sides-- which we've already
established as being equal-- we see that it's sitting
inside of the circle. So why don't we call
this an incircle? So circle I. Remember,
you label circles usually with the point at the center. Circle I is the incircle
of triangle ABC. And of course, the
radius of circle I-- so we could call this length r. We say r is equal
to IF, which is equal to IH, which
is equal to IG. We can call that
length the inradius. And it makes sense
because it's inside. When we were talking
about the intersection of the perpendicular bisectors,
we had our circumcenter because that was the
center of a circle that is circumscribed
about the triangle. Now, we're taking
the intersection of the angle bisectors. And then, using that, we're
able to define a circle that is kind of within the
triangle and whose sides are tangent to the circle. And since it's inside it,
we call this an incircle. We call the intersection of the
angle bisectors the incenter. And we call this distance
right over here the inradius.