Analytic Geometry

Distance Formula - The distance between two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ is given by the formula $d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$
Midpoint Formula - The midpoint $M$ of the line segment joining the points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ is $M = (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})$

Equation of a line

$m$ is the slope of the line

	Equation of a line
Two-point form	$y - y_{1} = \frac{y _{2} - y _{1}}{x _{2} - x _{1}} (x - x_{1})$	$(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ are any two points of the line, (if $x_{1} \neq = x_{2}$ then $m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}} = tan (θ)$ is the slope)
Slope–intercept form (y)	$y = m x + y_{0}$	assuming non-vertical line. $y_{0}$ is y-intercept
Slope–intercept form (x)	$y = m (x - x_{0})$	assuming non-horizontal line. $x_{0}$ is x-intercept
Standard Form	$A x + B y = C$
Point–slope form	$y = m (x - x_{0}) + y_{0}$	$(x_{0}, y_{0})$ is any point of the line. (assuming non-vertical line)
Intercept form	$\frac{x}{x _{0}} + \frac{y}{y _{0}} = 1$	assuming the line is not parallel to an axis and does not pass through the origin. The intercept values $x_{0}$ and $y_{0}$ are nonzero

Angle Between Two Lines - If two nonperpendicular lines have slopes $m_{1}$ and $m_{2}$ , then the tangent of the angle between the two lines is $tan θ = ∣ \frac{m _{2} - m _{1}}{1 + m _{1} m _{2}} ∣$
If two nonvertical lines are parallel, then their slopes are equal.
- Alternative Form: $ℓ_{1} ∥ ℓ_{2} ⟺ m_{1} = m_{2}$
If two lines (neither horizontal nor vertical) are perpendicular, then the product of their slopes is $- 1$ .
- Alternative Form: $ℓ_{1} ⊥ ℓ_{2} ⟺ m_{1} m_{2} = - 1$ where neither $ℓ_{1}$ nor $ℓ_{2}$ is a vertical line or a horizontal line.