Incenter and incircles of a triangle

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 to see what happens when we apply some of those ideas to triangles or the angles and 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 into 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 I could call this point D and then let me draw another angle bisector the one that bisects angle a BC so let me just draw this one looks it might look something like that right over there and I could maybe call this point e so ad bisects angle BAC and B e 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 a be e must be equal to the measure of angle e BC e BC now we see clearly that they have intersected at a point inside of the triangle right over there so let's call that point let me call it I just for fun I'm skipping a few letters but it'll be it's it's a useful letter that based on what we are going to call this in very short order and there's some interesting things 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 a D 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 a D we know that these two distances are going to be the same assuming this is this is the shortest distance between I and the sides and we also show 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 we could even let's label these some 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 I F we know that I F is going to be equal to is going to be equal to I G fair enough now I also sits on this angle bisector it also sits on B II which says that it must be equidistant from a the distance to a B must be the same as eyes distance to BC eyes distance to a B 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 draw 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 IG must be equal to I age but I F is also equal to IG so we can also say that I F I mean if I F is equal to I G is equal to IH we also know that if' I F 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 we this right here tells us tells us that I must be on angle bisector I is on angle bisector angle bisector sector of angle a see because it's equidistant to the two sides of angle a çb and what we've just shown is 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 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 eyes on the angle bisector of ACB so the angle the bisector of ACB will look something like this 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 draw it 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 in Center the in Center the in center of triangle ABC a BC and you're going to see the second why it's called the in-center when we talked about the circumcenter that was the center of a circle that was that could be circumscribed about the triangle I will see in about five seconds is the center of a circle that could be that is circumcised 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 the radius that has a radius equal to I f IG or I H well then you're going to have a circle that looks something like you're going to have a circle that looks something like this you're going to let me draw it a little bit better than that I don't have really 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 in 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 in circle in circle of triangle ABC and of course the radius of circle I so we could call call this length R you say R is equal to i F which is equal to I H which is equal to IG is equal to IG we can call that length the in radius and it makes sense because it's inside when we had 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 that whose sides are tangent to the circle and since it's inside it we call this an in circle we call that intersection of the angle bisectors the in center and we call this distance right over here the in radius