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## Geometry (all content)

### Course: Geometry (all content)>Unit 4

Lesson 5: Angle bisectors

# 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.

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• @ if 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..
• Same issue here but here's an explanation. Please refer to the diagram @

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.
• 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?
• 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.
• Is this always true? Or is it only true for most examples?
• 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.
• What is the definition of an incenter?
• 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
• 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). At , 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.
• What do the dotted lines that go between A&D, F&B, G&C and meet at I stand for? At & ?
(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! :)
• In , 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)