Topic 5: Special Relativity - derivation of Lorentz transformations

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I really want to understand how Lorentz Transformations come about!
I have analyzed the derivation of Lorentz Transformations (LT) in

[3] The special and general theory of relativity; Albert Einstein; The first part; About the special theory of relativity; 2006; swedish
[7] Modern Physics; Second edition; Randy Harris; Chapter 2; Special Relativity; 2008

I believe that the derivation in these two books is not self-consistent. Why?

I explain below and would like help with this. Both in [3] and [7] one uses three special cases to derive LT.
You start from two linear equations (I use my own notations to more easily refer to different parts of the derivation):
LEx': x' = Ax + Bt
LEt': t' = Cx + Dt
To determine the constants A, B, C, D, three special cases are used:
SC1: x' = 0, x = vt
SC2: x = 0, x' = -vt'
SC3: x' = ct', x = ct

We look at the first one that I depict in Fig. SC1.

SC1: x' = 0, x = vt

My comment:
At the time (t' > 0), an event occurs in the S'-origo (x' = 0).
The literature states that the distance between S and S' is x = vt.

I think this is wrong: It should be x = vt'.
Why?
S has no knowledge that the event occurred and where somewhere it happened!
S' instead knows that the event occurred when its clock showed time t'.
Then S' can definitely say that the distance between S and S' is vt'!

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Moderator Note

You have had multiple threads on this already. You clearly do not want to understand. Unless you are willing to listen to explanations and learn something, there is no point opening more threads.