Math Calculators, Lessons and Formulas

It is time to solve your math problem

mathportal.org

You included 0 formulas in your custom cheat sheet.

Cheat Sheet options

## How to generate cheat sheet ?

To create cheat sheet first you need to select formulas which you want to include in it. To select formula click at picture next to formula.

You can choose formulas from different pages.

After you have selected all the formulas which you would like to include in cheat sheet, click the "Generate PDF" button.

# Math Formulas: 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 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:

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

### 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:

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