The formula for the radius of the circle circumscribed about a triangle (circumcircle) is given by
 

where A_t is the area of the inscribed triangle.
 

The radius of incircle is given by the formula
 

where A_t = area of the triangle and s = semi-perimeter.
 

This page will define the following: incenter, circumcenter, orthocenter, centroid, and Euler line.
 

Incenter
Incenter is the center of the inscribed circle (incircle) of the triangle, it is the point of intersection of the angle bisectors of the triangle.
 

The radius of incircle is given by the formula

where A_t = area of the triangle and s = ½ (a + b + c). See the derivation of formula for radius of incircle.
 

Side
Side of a triangle is a line segment that connects two vertices. Triangle has three sides, it is denoted by a, b, and c in the figure below.
 

Vertex
Vertex is the point of intersection of two sides of triangle. The three vertices of the triangle are denoted by A, B, and C in the figure below. Notice that the opposite of vertex A is side a, opposite to vertex B is side B, and opposite to vertex C is side c.
 

The following are the formulas for cosine law for any triangles with sides a, b, c and angles A, B, C, respectively.
 

For any triangles with vertex angles and corresponding opposite sides are A, B, C and a, b, c, respectively, the sine law is given by the formula...
 

Half angle formulas can be derived from the double angle formulas, particularly, the cosine of double angle. For easy reference, the cosines of double angle are listed below:
 

Note that the equations above are identities, meaning, the equations are true for any value of the variable θ. The key on the derivation is to substitute θ with ½ θ.
 

The derivation of basic identities can be done easily by using the functions of a right triangle. For easy reference, these trigonometric functions are listed below.

a/c = sin θ
b/c = cos θ
a/b = tan θ
c/a = csc θ
c/b = sec θ
b/a = cot θ

From the right triangle shown below,
 

the trigonometric functions of angle θ are defined as follows:

Basic Identities
See the derivation of basic identities.

1. $\sin \theta = \dfrac{1}{\csc \theta} ~ \Leftrightarrow ~ \csc \theta = \dfrac{1}{\sin \theta}$
    
2. $\cos \theta = \dfrac{1}{\sec \theta} ~ \Leftrightarrow ~ \sec \theta = \dfrac{1}{\cos \theta}$
    
3. $\tan \theta = \dfrac{\sin \theta}{\cos \theta} = \dfrac{1}{\cot \theta} ~ \Leftrightarrow ~ \cot \theta = \dfrac{\cos \theta}{\sin \theta} = \dfrac{1}{\tan \theta}$