What is the incenter?

How the incenter is located

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.

Once the incenter is located, those equal perpendicular distances give the radius of the incircle. The circle touches all three sides because each side is the same distance from the center. This is why angle-bisector work and circle-fitting work meet at the same point.

The incenter is always inside because the three internal angle bisectors meet in the shared interior of the triangle. That makes it different from the circumcenter and orthocenter, whose positions can move outside when the triangle becomes obtuse.

• 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 the incenter helps

• 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.

Three incenter checks

• 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.