Incenter of a Triangle

What is the Incenter of a Triangle?
The incenter of a triangle is one of the four classical triangle centers, along with the orthocenter, centroid, and circumcenter.

The point of concurrency of the three angle bisectors of a triangle is the incenter. It is the center of the circle that can be inscribed in the triangle, making the incenter equidistant from the three sides of the triangle. The incenter is always located within the triangle.

The following diagram shows the incenter of a triangle. Scroll down the page for more examples and solutions on the incenters of triangles.

 

Geometry Worksheets
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Printable & Online Geometry Worksheets

What is an Angle Bisector?
An angle bisector of a triangle is a line segment (or ray) that divides the interior angle at a vertex into two equal angles.
Every triangle has three angle bisectors, one for each interior angle.

How to construct the Incenter?

1. Construct two angle bisectors for two of the interior angles.
   
2. The point where they intersect is the incenter.

Key Properties and Significance
The incenter is significant because it is the center of the triangle’s inscribed circle (or incircle).

1. Equidistant from Sides: The incenter is equidistant from all three sides of the triangle. This distance is the inradius (r) of the incircle. This means if you place the point of a compass on the incenter and open it to the perpendicular distance to any side, you can draw a circle that is tangent to all three sides of the triangle.
2. Always Inside: Unlike the orthocenter and circumcenter, the incenter of a triangle always lies inside the triangle, regardless of whether the triangle is acute, right, or obtuse. This is because it’s formed by the interior angle bisectors.
3. Unique Incircle: Every triangle has exactly one incircle and, therefore, exactly one incenter.

Incenter
This video demonstrates how to construct an incenter and inscribed circle using a compass and straight-edge.

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Incenter

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Construct the Incenter of a Triangle

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Constructing the Incenter

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