Angle bisectors
Incenter and incircles of a triangle Using angle bisectors to find the incenter and incircle of a triangle
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- I have triangle ABC here
- And in the last that we did started to explore
- some of the properties of points that are on angle bisectors
- And now why I wanna do this with you
- is just to 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 B, A, C
- and let me draw an angle bisector
- So the angle bisector might look something
- And I wanna 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 ABC
- So let me just draw this one
- It looks it might look something like that right over there
- And I can 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
- 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
- Let me call that I just for fun
- I'm skipping a few letters,
- but it will be it's a useful letter that based on
- what we are going to call this in very short order
- 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 2 sides of that angle
- So for example, I sits on AD so it's going to be equidistant
- from the 2 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 2 distances are going to be the same
- assuming this is the shortest distance between I and the sides
- And we've also shown in the 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 draw a perpendicular
- So that's why I drew the perpendiculars right over there
- And we could even let's label this some
- This is 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, we know that IF is going to be equal to,
- 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 from A
- the distance to AB must be the same as I is distanced 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 draw, if I draw 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 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, IF is equal to IH
- Pretty much common sense!
- If this is equal to that, that is equal to that,
- then these 2 have to be equal to each other
- But if I is equidistant from 2 sides of an angle,
- this is the 2nd part of what we proved in the previous video
- If you have a point that is equidistant from 2 sides of an angle,
- then that point must sit on the angle bisector for that angle
- So with this right here tells us,
- tells us that I must be on angle bisector
- I is on angle bisector, angle bisector sector of angle ACB
- 'cause it's equidistant to these 2 sides of angle ACB
- And what we have just shown is,
- is that there's a unique point inside the triangle
- that sits on all 3 angle bisectors
- It's not always obvious that if you took 3 lines,
- in fact normally, if you took 3 lines
- are not going to intersect in one point
- 2 lines is very reasonable thing to do,
- but 3 lines not always gonna intersects in one point,
- but once again, let me 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 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 have just shown that if you draw
- If you take the 3 angle bisectors of a triangle,
- it will intersect in a unique point right over there
- that sits on all 3 of them so would
- It seems worthwhile that we should call this something special
- And we do!
- And it's why I called it I
- We call I the incenter, the incenter
- The incenter of triangle ABC, ABC
- And you're gonna see the 2nd why it's called the incenter
- When we talk about the circumcenter,
- that was the center of a circle that was
- that could be circumscribed about the triangle
- I, we'll see in about 5 seconds,
- is the center of a circle that could be
- that is circums that is
- that can be put inside the triangle
- that's tangent to the 3 sides
- And how do we construct that?
- Well, we have 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 the center
- that has a radius equal to the distance
- between I and anyone of the sides which is equal,
- that has a radius, that has a radius equal to IF, IG or IH?
- 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
- Let me draw it a little bit better than that
- I don't have
- Rarely, 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 they're 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, incircle of triangle ABC
- And of course, the radius of circle I so we could call,
- we could call this length R
- Let's say R is equal to IF which is equal to IH
- which is equal to IG, is equal to IG
- We can call that length the inradius
- And it make 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 incircle
- We call the intersection of the angle bisectors the incenter
- And we call this distance right over here the inradius
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