Two - dimensional coordinate system
We found 12 formulas for two - dimensional coordinate system
1.
Distance between two points
$d=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}$
2.
Dividing line segment in ratio λ : 1
$x_0=\dfrac{x_1+\lambda x_2}{1+\lambda},\,\,\,\, y_0=\dfrac{y_1+\lambda y_2}{1+\lambda}$
3.
Midpoint of a line segment
$x_0=\dfrac{x_1+x_2}{2}, \,\,\, y_0=\dfrac{y_1+y_2}{2}$
4.
Centroid of a triangle
$x_0=\dfrac{x_1+x_2+x_3}{3},\,\,\, y_0=\dfrac{y_1+y_2+y_3}{3}$
5.
Incenter of a triangle
$x_0=\dfrac{ax_1+bx_2+cx_3}{a+b+c}, \, y_0=\dfrac{ay_1+by_2+cy_3}{a+b+c}, \\[1.4em] \text{ where } a=BC, b=CA, c=AB$
6.
Circumcenter of a triangle
$(x , y) = \left( ~ \frac{\begin{vmatrix} x_1^2+y_1^2 & y_1 & 1 \\ x_2^2+y_2^2 & y_2 & 1 \\ x_3^2+y_3^2 & y_3 & 1 \\ \end{vmatrix}} {2 \cdot \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \\ \end{vmatrix}}~,~ \frac{\begin{vmatrix} x_1 & x_1^2+y_1^2 & 1 \\ x_2 & x_2^2+y_2^2 & 1 \\ x_3 & x_3^2+y_3^2 & 1 \\ \end{vmatrix}} {2 \cdot \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \\ \end{vmatrix}}~ \right)$
7.
Orthocenter of a triangle
$(x , y) = \left( ~ \frac{\begin{vmatrix} y_1 & x_2x_3+y_1^2 & 1 \\ y_2 & x_3x_1 + y_2^2 & 1 \\ y_3 & x_1x_2+y_3^2 & 1 \\ \end{vmatrix}} {2 \cdot \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \\ \end{vmatrix}}~,~ \frac{\begin{vmatrix} x_1^2+y_2y_3 & x_1 & 1 \\ x_2^2+y_3y_1 & x_2 & 1 \\ x_3^2+y_1y_2 & x_3 & 1 \\ \end{vmatrix}} {2 \cdot \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \\ \end{vmatrix}}~ \right)$
8.
Area of a triangle
$A = \dfrac{1}{2} \begin{vmatrix} x_1 & y_1 & 1 \\[0.3em] x_2 & y_2 & 1 \\[0.3em] x_3 & y_3 & 1 \end{vmatrix}$
9.
Area of quadrilateral
$A=\dfrac{1}{2} \left[(x_1-x_2)(y_1+y_2)+(x_2-x_3)(y_2+y_3)+(x_3-x_4)(y_3+y_4)+(x_4-x_1)(y_4+y_1) \right]$
10.
Distance in polar coordinates
$d=\sqrt{r_1^2+r_2^2-2r_1r_2\cos\left(\phi_2-\phi_1\right)}$
11.
Rectangular coordinates to polar
$r=\sqrt{x^2+y^2} ,\, \phi= \tan\dfrac{y}{x}$