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

Centroid Located at intersection of the medians Properties: 2b c 2a * The centroid is always inside the triangle. The centroid of a triangle is a the point through which all the mass of a triangular plate seems to act. b Also known as its 'center of gravity' , 'center of mass' , or barycenter. * Each median divides the triangle into two smaller triangles of equal 2c 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.

½ s s ½ s

s s 3 x  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