Coordinate Geometry: Points & Lines

At a glance 12 formulas

1. 1Distance between two points$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$\text{points } (x_1, y_1) \text{ and } (x_2, y_2)$
2. 2Section formula$P = \left(\dfrac{mx_2 + nx_1}{m + n},\ \dfrac{my_2 + ny_1}{m + n}\right)$$P \text{ splits } (x_1, y_1) \text{ to } (x_2, y_2) \text{ in } m : n$
3. 3Midpoint M, centroid G$G = \left(\tfrac{x_1 + x_2 + x_3}{3},\ \tfrac{y_1 + y_2 + y_3}{3}\right)$$M = \left(\tfrac{x_1 + x_2}{2}, \tfrac{y_1 + y_2}{2}\right);\ \text{midpoints: same } G$
4. 4Triangle from its corners$\tfrac{1}{2}|ad - bc|\ \text{for } (0, 0), (a, b), (c, d)$$\text{shift any corner to } (0, 0) \text{ first}$
5. 5Slope$m = \dfrac{y_2 - y_1}{x_2 - x_1} = \tan\theta$$\theta \text{ with } x\text{-axis};\ ax + by + c = 0:\ m = -\tfrac{a}{b}$
6. 6Equation of a line$y - y_1 = m(x - x_1)$$\text{intercepts } a \text{ and } b:\ \dfrac{x}{a} + \dfrac{y}{b} = 1$
7. 7Parallel and perpendicular$\parallel:\ m_1 = m_2;\ \perp:\ m_1 m_2 = -1$$\perp \text{ to } ax + by = c:\ bx - ay = k$
8. 8Angle between two lines$\tan\theta = \left|\dfrac{m_1 - m_2}{1 + m_1 m_2}\right|$$\theta \text{ acute};\ 1 + m_1 m_2 = 0 \Rightarrow \theta = 90^\circ$
9. 9Distance to a line$d = \dfrac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}}$$\text{parallel, same } a, b:\ \dfrac{|c_1 - c_2|}{\sqrt{a^2 + b^2}}$
10. 10Three lines, no triangleThree lines fail to make a triangle if two are parallel (equal slopes) or all three pass through one point: solve two of them and put that point into the third.
11. 11Smallest cut-off triangleA line through $(h, k)$, with $h, k > 0$, cuts off at least area $2hk$ with the positive axes; the least comes when $(h, k)$ is the midpoint of the cut-off part.
12. 12Shortest path by reflectionTo go from A to a line and on to B (same side), reflect A in the line and join the image to B. In the x-axis $(x, y)$ goes to $(x, -y)$; in $y = x$, swap the coordinates.