Triangle Incenter definition - Math Open Reference

The Incenter of a triangle

Latin: in - "inside, within" centrum - "center"

The point where the three angle bisectors of a triangle meet.
One of a triangle's points of concurrency.

Try this Drag the orange dots on each vertex to reshape the triangle. Note the way the three angle bisectors always meet at the incenter.

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One of several centers the triangle can have, the incenter is the point where the angle bisectors intersect. The incenter is also the center of the triangle's incircle - the largest circle that will fit inside the triangle.

Properties of the incenter

Center of the incircle	The incenter is the center of the triangle's incircle, the largest circle that will fit inside the triangle and touch all three sides. See Incircle of a Triangle.
Always inside the triangle	The triangle's incenter is always inside the triangle. Adjust the triangle above by dragging any vertex and see that it will never go outside the triangle

Finding the incenter of a triangle

It is possible to find the incenter of a triangle using a compass and straightedge. See Constructing the the incircle of a triangle where finding the incenter is a step to constructing the incircle.

Summary of triangle centers

There are many types of triangle centers. Below are four of the most common.

Incenter		Located at intersection of the angle bisectors. See Triangle incenter definition
Circumcenter		Located at intersection of the perpendicular bisectors of the sides. See Triangle circumcenter definition
Centroid		Located at intersection of medians. See Centroid of a triangle
Orthocenter		Located at intersection of the altitudes of the triangle. See Orthocenter of a triangle

In the case of an equilateral triangle, all four of the above centers occur at the same point.

Summary of triangle centers

Related triangle topics

General

Perimeter / Area

Triangle types

Triangle centers

Congruence and Similarity

Solving triangles

Triangle quizzes and exercises