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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.