Coordinate Geometry

Gradient

$m=\dfrac{y_1-y_2}{x_1-x_2}$

Equation

$y-y_1=m(x-x_1)$
$y=mx+c$

Midpoint

$\left(\dfrac{x_1+x_2}{2}, \dfrac{y_1+y_2}{2}\right)$

Parallel Lines

$m_1=m_2$

Perpendicular Lines

$m_1=-\dfrac{1}{m_2}$
$m_1\times m_2=-1$

Area Of Quadrilateral

$A=\frac{1}{2}\left| \begin{array}{ccccc} x_1 & x_2 & x_3 & x_4 & x_1\\ y_1& y_2 & y_3 & y_4 & y_1\end{array} \right|$
$=\frac{1}{2}|(x_1y_2+x_2y_3+x_3y_4+x_4y_1)$
$-(x_2y_1+x_3y_2+x_4y_3+x_1y_4)|$
Note: coordinates should be in anti-clockwise direction

Circle

$(x-a)^2+(y-b)^2=r^2$
$(a,b)$ : centre of circle
$r$ : radius

Circle (Second Formula)

$x^2+y^2+2gx+2fy+c=0$
$(-g,-f)$ : centre of circle
$\sqrt{f^2+g^2-c}$ : radius