Geometry (all content)
Incenter and incircles of a triangle
The incenter of a triangle is the point at which the three angle bisectors intersect. To locate the incenter, one can draw each of the three angle bisectors, and then determine the point at which they all intersect. The incenter is also notable for being the center of the largest possible inscribed circle within the triangle. Created by Sal Khan.
Want to join the conversation?
- @3:12if the angle bisectors divide the angle into two equal parts, don't they intersect the opposite side of the triangle at the midpoint? (So D is the midpoint of BC?) In which case, isn't the shortest distance from the incenter also the midpoint? I was expecting the perpendicular drawn from the incenter to overlap the angle bisector at ID. Maybe I'm confusing everything..(4 votes)
- Same issue here but here's an explanation. Please refer to the diagram @3:12
I'm afraid the previous explanation was wrong and I have to change it.
We will proceed from "Angle Bisector Theorem"
The angle bisector theorem is TRUE for all triangles
In the above case, line AD is the angle bisector of angle BAC.
If so, the "angle bisector theorem" states that DC/AC = DB/AB
If the triangle ABC is isosceles such that AC = AB then DC/AC = DB/AB when DB = DC.
Conclusion: If ABC is an isosceles triangle(also equilateral triangle) D is the midpoint of BC then the angle bisector theorem is true.
However, if the triangle ABC is scalene such that AC ≠ AB then DC/AC ≠ DB/AB when DB = DC.
Conclusion: If the triangle ABC is scalene and D is the midpoint of BC then the angle bisector theorem is false.
This is a contradiction(that the angle bisector theorem is false).
Either the theorem is false or the assumption DB = DC is false.
The theorem is true(proven).
Therefore, DB = DC is false.
In conclusion, the angle bisector in isosceles triangles(for the angle between the equal sides) and equilateral triangles(for all angles) meet the opposite side at their midpoint.
For scalene triangles this CANNOT be the case.(6 votes)
- What's the difference between a centroid and the incenter? I know that the centroid is the point of intersection of the medians, and the incenter is the intersection of the angle bisectors, but don't the angle bisectors form the medians?(2 votes)
- In the case of a equilateral triangle, the point of intersection of the medians and angle bisectors are the same. If it's not equilateral, then they will be in different spots. Try it with a scalene triangle. The angle bisector of a side will not intersect in the same spot as the median of the other side.(2 votes)
- Is this always true? Or is it only true for most examples?(2 votes)
- I don't exactly know what you are wanting to know what is always true, but the things Sal said about the incircle is always true. You can find and incircle of any triangle even if it is not a right triangle.(2 votes)
- What is the definition of an incenter?(2 votes)
- The incenter is the point where all of the angle bisectors meet in the triangle, like in the video. It is not necessarily the center of the triangle.(1 vote)
- This might be out of context but what is the formula for coordinates of excentre?
could you please derive it with section formula or any other method .Thanks(2 votes)
- That means that IB=IC=IA right?(1 vote)
- No, I is the same distance from the three sides of the triangle (IF = IG = IH). At7:30, Sal reminds us that it is the circumcenter that is the same distance from the three vertices. For general triangles, those will be two different points.(3 votes)
- What do the dotted lines that go between A&D, F&B, G&C and meet at I stand for? At00:41&5:03?(1 vote)
- The line that goes through A&D is called a bisector, which pretty much just means that it cuts the triangle perfectly in half. there really isn't a line that goes through the points at F&B, but there are lines that are lines that go through F&I and E&B. E&B is another interceptor, it just cuts the circle in half in another direction. There is also no line that passes through G&C, though there is a line that passes through G&I and a line that passes through C that Sal leaves undefined on the other side. The line that goes through C is also a bisector. The lines that go through H&I, G&I and F&I are lines that are perpendicular to (or form right angles with) the sides of the triangle and all meet at the same point in the triangle. This point, point I, is then used as the center of the in-circle, which is just the circle that is drawn inside of the triangle. Hope this was helpful! And if you have any questions or clarifications you wish to make about my answer, please, feel free to do so! :)(2 votes)
- In0:30, it talks about an angle bisector. What is an angle bisector?(1 vote)
- It is a line segment which divides an angle into two equal halves.(1 vote)
- which is the one who has marks incenter or circumcenter? and I'm confused of centroid rules(1 vote)
- What is the difference between the circumcenter, incenter, and incircle?.(1 vote)
- The circumcenter is where the three perpendicular bisectors intersect, and the incenter is where the three angle bisectors intersect. The incircle is the circle that is inscribed inside the triangle. Its center is the incenter.(1 vote)
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.