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 the triangle and construct its bisector. Pick any one angle of a triangle and construct its bisector.
2 Pick one of the remaining angles of the triangle and construct its bisector. Pick one of the remaining angles of a triangle and construct its bisector.
3 Mark the intersection of the two lines as the incenter. Mark the intersection of the two lines as the incenter.
4 Construct a line perpendicular to any side through the incenter. 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. 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 A regular hexagon 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 A regular hexagon with the perpendicular bisector of one of the sides drawn in. Draw the perpendicular bisector of any side.
3 A regular hexagon with the perpendicular bisector of two of the non-opposite sides drawn in. Draw the perpendicular bisector of any other side that is not opposite the side you used in step 2.
4 A regular hexagon with the perpendicular bisector of two of the non-opposite sides drawn in. The intersection of the two perpendicular bisectors in labeled 'center'. Label the intersection of the two perpendicular bisectors as 'center'.
5 A regular hexagon with the perpendicular bisector of two of the non-opposite sides drawn in. The intersection of the two perpendicular bisectors in labeled 'center'. A circle is drawn with the center at 'center' and the edge at the midpoint of one of the sides. 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

More Information

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

Cite this article as:

McAdams, David E. Incenter. 8/7/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/i/incenter.html.

Image Credits

Revision History

8/6/2018: Removed broken links, updated license, implemented new markup, implemented new Geogebra protocol. (McAdams, David E.)
10/23/2010: Changed article to apply to incenters of a polygon, rather than just a triangle. Added section on constructing the incenter of a regular polygon. (McAdams, David E.)
2/11/2010: Added "References". (McAdams, David E.)
11/18/2008: Changed manipulative to GeoGebra. (McAdams, David E.)
8/24/2007: Simplified figure 1, added reference to triangle article, added incircle. (McAdams, David E.)
7/30/2007: Initial version. (McAdams, David E.)

All Math Words Encyclopedia is a service of Life is a Story Problem LLC.
Copyright © 2018 Life is a Story Problem LLC. All rights reserved.
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License