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How many techniques are there to test coliniariry of n points?

For example, suppose we have 4 points A, B , C, D. How many ways can it be tested that they are collinear?

This answer lists 03 techniques: https://math.stackexchange.com/a/221530/6704

isn't there another technique like the following: To test whether enter image description here

are equal?

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Perhaps along the lines of what you are thinking, if you subtract one point from the rest so that one point is $(0,0)$, then all the points are on a common line if and only if every point $(x_i,y_i)$ is an exact multiple of every other point. As long as the line is not horizontal, you can test this for example by computing $x_i/y_i$ for each remaining point (after subtracting one point from all the rest) and checking that this "slope" is equal for all your remaining points.

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