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
Step  Illustration  Discussion 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 

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.)