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Angle Bisectors and Incenters

In geometry, the bisector of an angle of a triangle is a line segment that bisects the angle, that is, divides it into two equal angles. In a triangle, the bisector of an angle is the line through the vertex that divides the opposite side into segments with lengths proportional to the adjacent sides. In a circle, the bisector of an angle is a chord that passes through the center of the circle and divides the angle in half. The incenter of a triangle is the center of its incircle, the circle passing through all three vertices of the triangle.


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Video Transcript

Okay. Hi. We're back with even Mawr special segments, lions and raise regarding triangles. And today we're gonna talk about angle by sectors and the whole lot of stuff going on with angle by sectors of a triangle. If you have a triangle here, okay, you can have or can construct what we call an angle by sector. That's a ray or that does it as an end point at one Vertex of the triangle and then basically cuts the angle of the angle. It cuts the angle in two pieces. So this angle here in this angle are now congruent. So, you know, already questions I could ask you I could give you, You know, I could tell you the entire angle and ask you for what half of it is. I could do that with numerical values I could do with algebraic expressions. And then you could set one equal to half the other or double one, able to the whole or one equal to the other, depending on what expressions are labeled where. But what? An angle by sector is essentially a ray who has endpoint on one vertex of the triangle and then proceeds that cut the angle from which it is originating from, and then at some point, crosses through the opposite side of the triangle by no means across through the midpoint of the opposite side. Because there's nothing to do. It's not a median. There's nothing to do with bisecting. The opposite side has everything to do with bisecting the angle from which it originates from. That's an angle by sector couple special things. One. Any point on the by sector is going to be equally distant to the two sides that form the angle so that those two distances air equal. And so if you have an angle bio sector, you then have any point on the angle. Bio sector equi distant to the two sides. They formed the angle. But also the converse of that term is if you have and know that this point happens to be equal distant from either side. Did you know for a fact it's an angle by sector? Okay, which means we have a bike conditional situation here. The situation is, if you have an angle by sector than any point on, it is equidistant to the two sides that forms the angle, or if you know you have a point that's equidistant two sides that forms have given angle. That ray must be an angle by sector. Now, like the medians of a triangle, there are three angle by sectors. They all crossed through the interior of the triangle so I could draw from the top Vertex this angle by sector and from this protects this angle by sector. And like everything else, like like the altitudes. And like the medians, all three of these angle by sectors are concurrent. They will meet at one very, very, very, very special point. And that point is known as Thean center. Yeah, okay. So so far to recap you have a triangle. You have array that originates from any Vertex of that triangle and then bisects the angle that it originates from it will cross to the opposite side of the triangle. All three angle by sectors will exist inside of the triangle. They will meet at one point of concurrency, which we called the center in center apartment. Now it's very special. If you go back to this whole, any point on an angle by sector is equidistant to the sides of the triangle. Then that incentive becomes very, very, very important because what we get is this very special point here in the middle. And it happens to be that its distance from each of the sides is equal, because if that point resides on all three of those angle by sectors than that point is, equidistant toe all three sides. And the reason why we call it an in center is because you can actually construct a circle inside of the triangle that's tangent to the triangle at those given points here, here and here, because by default, those three distances are congruent because of the relationship of any point on an angle by sector, with respect to the sides of the triangle. So regardless of the type of triangle we're dealing with, we could have a right triangle, and I could have that angle by sector. This thing let me do that one again. Yeah, that angle by sector and this angle by sector, and what we know is this point here is Thean center that distance that distance and it's not drawn to scale per saved. This distance are all equal, and what you get is a circle called the Encircle that's gonna be tangent. All three sides notice that unlike the altitudes But like the medians again, all the angle by sectors are within the triangle and that the point of concurrency stays inside of the triangle. So if you look at an obtuse triangle, the same thing happens. We have that angle by sector that ain't go by sector. That angle by sector point of concurrency is here. Yeah, it's equidistant to that side, that side and that side. Pardon me, that side because we're looking at the perpendicular distance and you can get a circle. Okay, senator, at the end center. So we have a lot of special relationships here. One, an angle by sector is very special because and let's draw an angle here for the time being. If you have an angle by a sector, any point on their do that again, any point on there is going to be one more time is going to be equally distant to the suicides that form the angle and vice versa. If we know that we have a point that happens to be eager, distant to the two sides, then we know that the ray it's on is an angle by sector. So that has nothing to do with the triangle per se, but everything to do with what an angle by sector is. Now throw it into a triangle. You have three of them. They all intersect at this point called the in center. And because of the relationship of the points on any angle by sector to the sides of the triangle, it's gonna be eagerness into all three sides, and you're gonna be ableto construct the circle known as Thean Circle, with the center at that point known as Thean Center. So thistles, basically a rundown of what a angle by sector is within a triangle and the special properties and characteristics that they have.