Detailed definition
Understanding Incenter
Incenter is the point where the three angle bisectors of a triangle intersect. The incenter is the point where the angle bisectors intersect. That point is also the center of the incircle, the circle tangent to all three sides of the triangle.
The incenter is always inside the triangle, no matter the triangle type. That makes it easier to track than some other centers whose positions depend on whether the triangle is acute, right, or obtuse.
This center matters because it turns equal-angle information into equal-distance-to-side information. The bisectors lead to a point whose perpendicular distances to the three sides are the same.
Key facts
Important ideas to remember
- The incenter is the point where the angle bisectors intersect.
- The incenter is formed by the intersection of the three angle bisectors.
- It is the center of the triangle's incircle.
- The incenter is always inside the triangle.
Where it is used
Where incenter shows up
- Use the incenter when working with angle bisectors and inscribed circles in triangles.
- Use it in constructions that require the incircle or equal distances to the sides.
- Use it in proofs where bisector concurrency gives information about tangency or distances.
Common mistakes
What to watch out for
- Do not confuse distance to the sides with distance to the vertices; that latter idea belongs to the circumcenter.
- Do not label the incenter from a guessed interior point without showing the bisectors that create it.
- Do not mix the incenter with the centroid or orthocenter just because all may lie inside some triangles.