Incenter

Pronunciation: /ˈɪnˌsɛn.tər/ Explain

 Click on the blue points to change the figure. Click on the check boxes to see the incircle and to see how to incenter is drawn. Manipulative 1 - Incenter of a Regular Polygon Created with GeoGebra.

The incenter of a polygon is a point that is the center of the circle that intersects each side of the polygon exactly once. The incircle of a triangle can be constructed by finding the intersection of the angle bisectors. The incircle of a regular polygon is located at the intersection of the perpendicular bisectors of the sides of the polygon.

The incircle of a geometric figure is the circle that is tangent to all the sides of a triangle. The incircle touches each of the sides exactly once.

How to Construct the Incenter and Incircle of a Triangle

 1 Pick any one angle of a triangle and construct its bisector. 2 Pick one of the remaining angles of a triangle and construct its bisector. 3 Mark the intersection of the two lines as the incenter. 4 Construct a line perpendicular to any side through the incenter. Mark the point where the line intersects the side to which it is perpendicular as point A. 5 Construct circle with the center at the incenter and the radius the distance from the incenter to point A.

How to Construct the Incenter and Incircle of a Regular Polygon

StepIllustrationDiscussion and Justification
1 The center of a regular polygon is at the point of concurrency of perpendicular bisectors of any two sides that are not opposite each other.
2 Draw the perpendicular bisector of any side.
3 Draw the perpendicular bisector of any other side that is not opposite the side you used in step 2.
4 Label the intersection of the two perpendicular bisectors as 'center'.
5 Draw a circle with the center at the point labeled 'center' and the edge where one of the perpendicular bisectors intersects a side.
Table 3 - How to construct the center and incircle of a regular polygon

1. McAdams, David E.. All Math Words Dictionary, incenter. 2nd Classroom edition 20150108-4799968. pg 95. Life is a Story Problem LLC. January 8, 2015. Buy the book

• McAdams, David E.. Center. allmathwords.org. Life is a Story Problem LLC. 3/12/2009. https://www.allmathwords.org/en/c/center.html.

McAdams, David E. Incenter. 4/23/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. https://www.allmathwords.org/en/i/incenter.html.

Revision History

4/23/2019: Updated equations and expressions to new format. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)