How to prove that three points are collinear

When the coordinates of three points are known
Method 1: take two points to establish a straight line
The analytical formula of the straight line is calculated
Take the coordinate of the third point as an example to see if it meets the analytical formula
Method 2: set three points as a, B and C
Using vector to prove: a times AB vector = AC vector (where a is non-zero real number)
Method 3: ab slope and AC slope were calculated by point difference method
Equality means that three points are collinear

How to prove that four points are coplanar and three points are collinear?
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Three points collinear - use two-point coordinates to get the linear equation of these two points, and see if the coordinates of another point conform to this equation
It's the same with coplanarity of four points - use the coordinates of three points to solve the equation of the surface determined by the three points, and see if the coordinates of another point conform to the equation

Prove that three points are collinear
Taking both sides AB and AC of the triangle ABC as the edge type, the square ABDE and acfg are made out, and then taking BC as the hypotenuse, the isosceles RT △ MBC are made out to the same side of the triangle ABC to prove that D, m and F are collinear

Make a vertical line to BC through D, m and F
P, Q, t are the vertical foot
We only need to prove DP + FQ = 2Mt = BC
Then go through a to make BC vertical line, and the vertical foot is h
Easy to know
Triangle DPB ≌ BHA, AHC ≌ CQF
So DP + FQ = BH + CH = BC