Chapter 7.2-7.3 Triangle Centers and Regular Polygon Properties
Some triangle centers – http://www.mathopenref.com/trianglecenters.html
(for a funny song, click on http://www.youtube.com/watch?v=Uzx8wI07M4E)
There are many types of triangle centers. Below are four common ones. There is a page for each one. Click on the link to probe
deeper. These centers are POINTS OF CONCURRENCY of some “Special Lines” in the triangle.
Incenter
Located at intersection of the angle bisectors.
Properties: The incenter is the center of the triangle's incircle, the
largest circle that will fit inside the triangle and touch all three sides.
See Incircle of a Triangle.
Always inside the triangle
The incenter is equidistant to the sides of the triangle
See Triangle incenter definition and
How to Construct the Incenter of a Triangle
Circumcenter
Located at intersection of the perpendicular bisectors of the sides .
Properties: The circumcenter is the center of a circle circumscribed
around the vertices of the triangle.
The circumcenter is equidistant to the vertices of the triangle
See Triangle circumcenter definition and
How to Construct the Circumcenter of a Triangle
2b c
2a
a
2c
b
Centroid
Located at intersection of the medians
Properties:
* The centroid is always inside the triangle. The centroid of a triangle is
the point through which all the mass of a triangular plate seems to act.
Also known as its 'center of gravity' , 'center of mass' , or barycenter.
* Each median divides the triangle into two smaller triangles of equal
area.
*The centroid is exactly two-thirds the way along each median. Put
another way, the centroid divides each median into two segments
whose lengths are in the ratio 2:1, with the longest one nearest the
vertex.
See Triangle centroid definition
and Constructing the Centroid of a Triangle.
Orthocenter
Located at intersection of the altitudes
See Triangle orthocenter definition
and Constructing the Orthocenter of a Triangle.
Chapter 7.2-7.3 Triangle Centers and Regular Polygon Properties
Incenter of a regular polygon
The point where the interior angle bisectors intersect.
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:
1. The incircle - the largest circle that will fit inside the polygon
2. The circumcircle - the circle that passes through every vertex.
Radius of a regular polygon (also Circumradius)
Definition: The distance from the center of a regular polygon to any vertex .
The radius of a regular polygon is the distance from the center to any vertex. It will be the same for any
vertex. The radius is also the radius of the polygon's circumcircle, which is the circle that passes through
every vertex. In this role, it is sometimes called the circumradius.
Irregular polygons are not usually thought of as having a center or radius.
Chapter 7.2-7.3 Triangle Centers and Regular Polygon Properties
Apothem of a Regular Polygon
Definition: A line segment from the center of a regular polygon to the midpoint of a side.
The apothem is also the radius of the incircle of the polygon. For a polygon of n sides, there are n possible apothems, all
the same length of course. The word apothem can refer to the line itself, or the length of that line. So you can correctly
say 'draw the apothem' and 'the apothem is 4cm'.
For a regular hexagon, the apothem can be found given the length of a side because the central angle formed by the
vertices of a hexagon is 60°. Because each angle is the same. the triangles formed are equalilateral triangles. Because the apothem
is the perpendicular bisector of an equilateral triangle it forms 2 30-60-90 triangles.
½s
s
½s
s
x
s 3
2
Chapter 7.2-7.3 Triangle Centers and Regular Polygon Properties
Incircle of a Polygon
Definition: The largest circle the will fit inside a polygon that touches every side
The incircle of a regular polygon is the largest circle that will fit inside the polygon and touch each side in
just one place (see figure above) and so each of the sides is a tangent to the incircle. If the number of
sides is 3, this is an equilateral triangle and its incircle is exactly the same as the one described in Incircle
of a Triangle.
The inradius of a regular polygon is exactly the same as its apothem. The formulas below are the same
as for the apothem.
Circumcircle of a Polygon
Definition: The circle that passes through each vertex of the regular polygon.
Also - circumscribed circle.
For Regular Polygons
The circumcircle of a regular polygon is the circle that passes through every vertex of the polygon. If the number of sides
is 3, then the result is an equilateral triangle and its circumcircle is exactly the same as the one described in Circumcircle
of a Triangle.
Chapter 7.2-7.3 Triangle Centers and Regular Polygon Properties
Finding the radius
The radius of a regular polygon is the distance from the center to any vertex. It will be the same for any vertex. The radius
is also the radius of the polygon's circumcircle, which is the circle that passes through every vertex. In this role, it is
sometimes called the circumradius.
Irregular polygons are not usually thought of as having a center or radius.
Irregular Polygons
Irregular polygons are not usually considered as having a circumcircle. If you draw a polygon at random, it is unlikely there
will be a circle that passes through every vertex. Irregular quadrilaterals that have a circumcircle are called cyclic
quadrilaterals. A convex quadrilateral is cyclic if and only if the four perpendicular bisectors to the sides are concurrent.
This common point is the circumcenter, A convex quadrilateral ABCD is cyclic if and only if its opposite angles are
supplementary