# Triangles in Two Dimensions

## Area of the triangle

The area of the triangle formed by the three lines:

\begin{aligned} A_1x + B_1y + C_1 &= 0 \\ A_2x + B_2y + C_2 &= 0 \\ A_3x + B_3y + C_3 &= 0 \end{aligned}

is given by

$$A = \frac{\begin{vmatrix} A_1 & B_1 & C_1 \\ A_2 & B_2 & C_2 \\ A_3 & B_3 & C_3 \end{vmatrix}^2} {2\cdot \begin{vmatrix} A_1 & B_1 \\ A_2 & B_2 \end{vmatrix} \cdot \begin{vmatrix} A_2 & B_2 \\ A_3 & B_3 \end{vmatrix} \cdot \begin{vmatrix} A_3 & B_3 \\ A_1 & B_1 \end{vmatrix}}$$

The area of a triangle whose vertices are $$P_1(x_1, y_1) , P_2(x_2, y_2)$$ and $$P_3(x_3, y_3)$$ is given by:

$$A = \frac{1}{2} \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix}$$

and by:

$$A = \frac{1}{2} \begin{vmatrix} x_2-x_1 & y_2-y_1 \\ x_3-x_1 & y_3-y_1 \end{vmatrix}$$

## Centroid

The centroid of a triangle whose vertices are $$P_1(x_1,y_1), P_2(x_2, y_2)$$ and $$P_3(x_3, y_3)$$ is given by:

$$(x,y) = \left( \frac{x_1+x_2+x_3}{3} , \frac{y_1+y_2+y_3}{3} \right)$$

## Incenter

The incenter of a triangle whose vertices are $$P_1(x_1,y_1), P_2(x_2, y_2)$$ and $$P_3(x_3, y_3)$$ is given by:

$$(x,y) = \left( \frac{a\,x_1+b\,x_2+c\,x_3}{3} , \frac{a\,y_1+b\,y_2+c\,y_3}{3} \right)$$

where a is the length of $$P_2P_3$$, b is the length of $$P_3P_1$$, and c is the lenght of $$P_1P_2$$

## Circumcenter

The circumcenter of a triangle whose vertices are $$P_1(x_1,y_1), P_2(x_2, y_2)$$ and $$P_3(x_3, y_3)$$ is given by:

$$(x , y) = \left( ~ \frac{\begin{vmatrix} x_1^2+y_1^2 & y_1 & 1 \\ x_2^2+y_2^2 & y_2 & 1 \\ x_3^2+y_3^2 & y_3 & 1 \\ \end{vmatrix}} {2 \cdot \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \\ \end{vmatrix}}~,~ \frac{\begin{vmatrix} x_1 & x_1^2+y_1^2 & 1 \\ x_2 & x_2^2+y_2^2 & 1 \\ x_3 & x_3^2+y_3^2 & 1 \\ \end{vmatrix}} {2 \cdot \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \\ \end{vmatrix}}~ \right)$$

## Orthocenter

The orthocenter of a triangle whose vertices are $$P_1(x_1,y_1), P_2(x_2, y_2)$$ and $$P_3(x_3, y_3)$$ is given by:

$$(x , y) = \left( ~ \frac{\begin{vmatrix} y_1 & x_2x_3+y_1^2 & 1 \\ y_2 & x_3x_1 + y_2^2 & 1 \\ y_3 & x_1x_2+y_3^2 & 1 \\ \end{vmatrix}} {2 \cdot \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \\ \end{vmatrix}}~,~ \frac{\begin{vmatrix} x_1^2+y_2y_3 & x_1 & 1 \\ x_2^2+y_3y_1 & x_2 & 1 \\ x_3^2+y_1y_2 & x_3 & 1 \\ \end{vmatrix}} {2 \cdot \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \\ \end{vmatrix}}~ \right)$$