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Triangles in two dimensions

Area of the triangle

The area of the triangle formed by the three lines:

A 1 x + B 1 y + C 1 A 2 x + B 2 y + C 2 A 3 x + B 3 y + C 3 = 0 = 0 = 0

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

is given by

A = ∣ ∣ ∣ ∣ A 1 A 2 A 3 B 1 B 2 B 3 C 1 C 2 C 3 ∣ ∣ ∣ ∣ 2 2 \cdot ∣ ∣ ∣ A 1 A 2 B 1 B 2 ∣ ∣ ∣ \cdot ∣ ∣ ∣ A 2 A 3 B 2 B 3 ∣ ∣ ∣ \cdot ∣ ∣ ∣ A 3 A 1 B 3 B 1 ∣ ∣ ∣

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

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 = 1 2 ∣ ∣ ∣ ∣ x 1 x 2 x 3 y 1 y 2 y 3 111 ∣ ∣ ∣ ∣

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

and by:

A = 1 2 ∣ ∣ ∣ x 2 - x 1 x 3 - x 1 y 2 - y 1 y 3 - y 1 ∣ ∣ ∣

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

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) = (x 1 + x 2 + x 3 3, y 1 + y 2 + y 3 3)

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

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) = (a x 1 + b x 2 + c x 3 3, a y 1 + b y 2 + c y 3 3)

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

where $a$ is the length of $P_2P_3$ , $b$ is the length of $P_3P_1$ , and $c$ is the length 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) = ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ∣ ∣ ∣ ∣ ∣ x 2 1 + y 2 1 x 2 2 + y 2 2 x 2 3 + y 2 3 y 1 y 2 y 3 1 1 1 ∣ ∣ ∣ ∣ ∣ 2 \cdot ∣ ∣ ∣ ∣ x 1 x 2 x 3 y 1 y 2 y 3 1 1 1 ∣ ∣ ∣ ∣, ∣ ∣ ∣ ∣ ∣ x 1 x 2 x 3 x 2 1 + y 2 1 x 2 2 + y 2 2 x 2 3 + y 2 3 1 1 1 ∣ ∣ ∣ ∣ ∣ 2 \cdot ∣ ∣ ∣ ∣ x 1 x 2 x 3 y 1 y 2 y 3 1 1 1 ∣ ∣ ∣ ∣ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟

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

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) = ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ∣ ∣ ∣ ∣ ∣ y 1 y 2 y 3 x 2 x 3 + y 2 1 x 3 x 1 + y 2 2 x 1 x 2 + y 2 3 1 1 1 ∣ ∣ ∣ ∣ ∣ 2 \cdot ∣ ∣ ∣ ∣ x 1 x 2 x 3 y 1 y 2 y 3 1 1 1 ∣ ∣ ∣ ∣, ∣ ∣ ∣ ∣ ∣ x 2 1 + y 2 y 3 x 2 2 + y 3 y 1 x 2 3 + y 1 y 2 x 1 x 2 x 3 1 1 1 ∣ ∣ ∣ ∣ ∣ 2 \cdot ∣ ∣ ∣ ∣ x 1 x 2 x 3 y 1 y 2 y 3 1 1 1 ∣ ∣ ∣ ∣ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟

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

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