How to prove that three points are collinear

How to prove that three points are collinear

If we prove that ABC three points are collinear,
1. Prove ∠ ABC = 180 degree
2. Prove that the line segment BA (or AB) and the line segment BC (or CB) are parallel, and because there is a common point, they are collinear
3. Prove that the vector BA (or AC) and the vector BC (or CB) are parallel and collinear because they have a common point (if you have studied vectors)

A proof of three points collinear
It is known that de in triangle ABC is parallel to BC, AP is the middle line of triangle ade, AQ is the middle line of triangle ABC. It is proved that a, P and Q are collinear
Because De is parallel to BC
It is easy to prove that ape is similar to AQC
De parallel BC can't prove that a triangle ape is similar to a triangle AQC. It can only prove that a diagonal is equal (angle AEP = angle ACQ). The others can't prove it at all, because you use the opposite method, or assume that three points are not collinear, so angle PAE doesn't = angle QAC Ah, then you can only prove that the two sides next to the equal angle are proportional, so you are wrong starting from line 7, or I don't understand, but thank you. If you answer well, add points

Because triangle ABC is similar to triangle ade, a is the position center
So the line between the midpoint of BC and the midpoint of de must pass through the quasi center a
The line of the corresponding point in the iconicity graph must pass through the iconicity center
Or take a as the origin, AB, AC as the XY axis to establish affine (oblique) coordinate system, and then find BC, de equation, and then use the midpoint formula to find PQ, prove that the PQ line must pass through the origin a, analytic geometry is too troublesome!

In space, how to prove that three points are collinear and three lines are collinear

Three point collinear: a belongs to L, B belongs to L, C belongs to L. three line power supply: a cross B = a, a belongs to C