Gradient of Parallel and Colinear Lines

If line segments \(AB\) and \(CD\) are parallel, their gradients are equal such that \(m_{AB} = m_{CD}\)
Given \( A = (x_1, y_1), B = (x_2, y_2), C = (x_3, y_3)\), and \(D = (x_4, y_4)\)

\( \frac{y_2 - y_1}{x_2 - x_1} = \frac{y_4 - y_3}{x_4 - x_3} \)

If three or more points lie on the same line segment, they are called colinear and all have an equal gradient
Given \(A \ (x_1, y_1), B \ (x_2, y_2)\) and \(C \ (x_3, y_3) \) all lie on the same straight line, then \(A, B,\) and \(C\) are colinear.

\(m_{AB} = m_{BC} = m_{AC} \)