Straight Lines

Distance, section and locus

Distance
$d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$
Section formula
$(\frac{m x _{2} + n x _{1}}{m + n}, \frac{m y _{2} + n y _{1}}{m + n})$ internal $;$ $m \to - m$ for external
Area of a triangle
$Δ = \frac{1}{2} x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2})$

Centroid
$G = (\frac{x _{1} + x _{2} + x _{3}}{3}, \frac{y _{1} + y _{2} + y _{3}}{3})$
Incentre
$I = (\frac{a x _{1} + b x _{2} + c x _{3}}{a + b + c}, \frac{a y _{1} + b y _{2} + c y _{3}}{a + b + c})$
Euler line
$O, G, H$ collinear $;$ $H G : GO = 2 : 1$
$O$ circumcentre, $H$ orthocentre.

$y = m x + c;$ $y - y_{1} = m (x - x_{1});$ $\frac{x}{a} + \frac{y}{b} = 1;$ $x cos α + y sin α = p$
Two points
$y - y_{1} = \frac{y _{2} - y _{1}}{x _{2} - x _{1}} (x - x_{1})$
Parametric
$\frac{x - x _{1}}{cos θ} = \frac{y - y _{1}}{sin θ} = r$

Angle between lines
$tan θ = \frac{m _{1} - m _{2}}{1 + m _{1} m _{2}};$ $∥: m_{1} = m_{2};$ $⊥: m_{1} m_{2} = - 1$
Point to line
$d = \frac{∣ a x _{1} + b y _{1} + c ∣}{a ^{2} + b ^{2}}$
Parallel lines
$d = \frac{∣ c _{1} - c _{2} ∣}{a ^{2} + b ^{2}}$
Foot and image
$\frac{x - x _{1}}{a} = \frac{y - y _{1}}{b} = - \frac{a x _{1} + b y _{1} + c}{a ^{2} + b ^{2}}$ (foot) $,$ $= - \frac{2 ( a x _{1} + b y _{1} + c )}{a ^{2} + b ^{2}}$ (image)
Same side
$P, Q$ on the same side of $a x + b y + c = 0 ⟺ (a x_{1} + b y_{1} + c) (a x_{2} + b y_{2} + c) > 0$

Family
$L_{1} + λ L_{2} = 0$ passes through $L_{1} \cap L_{2}$
Concurrency
$a_{1} a_{2} a_{3} b_{1} b_{2} b_{3} c_{1} c_{2} c_{3} = 0$
Angle bisectors
$\frac{a _{1} x + b _{1} y + c _{1}}{a _{1}^{2} + b _{1}^{2}} = \pm \frac{a _{2} x + b _{2} y + c _{2}}{a _{2}^{2} + b _{2}^{2}}$
Pair of lines
$a x^{2} + 2 h x y + b y^{2} = 0 : tan θ = \frac{2 h ^{2} - ab}{∣ a + b ∣};$ $⊥$ if $a + b = 0$
Represents a pair
$a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0 : ab c + 2 f g h - a f^{2} - b g^{2} - c h^{2} = 0$