

Incenter of a regular polygon
The point where the interior angle bisectors intersect.
Try this
Drag any orange dot and change the number of sides. Note where the bisectors of the interior angles meet.
If you
bisect
the interior angles
of a
regular polygon,
the bisectors will always converge at the same point  called the incenter of the polygon.
The incenter is also the center of:
 The incircle  the largest circle that will fit inside the polygon
 The circumcircle  the circle that passes through every vertex.
What about irregular polygons?
For irregular polygons  where the side lengths and interior angles are all different  the angle bisectors do not meet at a single point, so irregular polygons have no incenter.
Except triangles!
All triangles have an incenter, regardless of shape and proportions. See
Incenter of a Triangle.
Other centers
Regular polygons (and all triangles) have other centers too:
 Circumcenter  the point where the three perpendicular bisectors of the sides meet.
 Centroid  the point where the altitudes meet.
Note however that in the case of regular polygons these centers are all in the same place.
In the case of triangles they are usually in different places.
(In equilateral triangles, which are essentially threesided regular polygons, they are in the same place).
Other polygon topics
General
Types of polygon
Area of various polygon types
Perimeter of various polygon types
Angles associated with polygons
Named polygons
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