Centers of a Triangle

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.
 

incenter-incircle.jpg

 

The radius of incircle is given by the formula

$r = \dfrac{A_t}{s}$

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

Properties of Triangle

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.
 

Plane Geometry

Triangle

 

triangle.png

 

Perimeter, $P = a + b + c$

Semi-perimeter, $s = \frac{1}{2}P = \frac{1}{2}(a + b + c)$

Sum of included angles, $A + B + C = 180^\circ$

$\begin{align}
\text{Area, } A & = \frac{1}{2} bh \\
& = \frac{1}{2}ab \sin \theta \\
& = \sqrt{s(s - a)(s - b)(s - c)}
\end{align}$
 

Derivation of the Half Angle Formulas

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:
 

cos 2θ = 1 - 2sin2 θ   →   Equation (1)
cos 2θ = 2cos2 θ - 1   →   Equation (2)

 

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.