$$\text{Distance} = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$$

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

$$\text{Gradient} = \dfrac{y_1 - y_2}{x_1 - x_2} = \tan\theta$$

$$\text{Equation of line: } y - y_1 = m(x - x_1)$$

$$\text{Lines parallel} \iff m_1 = m_2$$

$$\text{Lines perpendicular} \iff m_1 m_2 = -1$$

$$\text{Area of polygon} = \dfrac{1}{2} \left| \begin{vmatrix} x_1 & x_2 & x_3 & \cdots & x_1 \\ y_1 & y_2 & y_3 & \cdots & y_1 \end{vmatrix} \right|$$

$$(x - a)^2 + (y - b)^2 = r^2$$

$$x^2 + y^2 - 2ax - 2by + c = 0$$