Original Question: How do you draw the clothesline? After a downtrend, when the uptrend resumes, where is the starting point? I refer here to the NDX clothesline when you mentioned 12900 …I did not understand why this is a point of importance.
You start drawing the clothesline once you have two significant price peaks which you can connect.
Here is an example:

The clothesline is a trend line on a chart with a logarithmic price scaling. Therefore a straight line represents an exponential growth. This corresponds to the typical expansion behavior of businesses in their hyper-growth phase. Consecutive constant quarterly EPS or sales percentage increases also represent exponential growth.
Stocks and indices oftentimes follow such a trend line loosely. Typically the price peaks reach exactly this line. The timespan between those visits can be anything from weeks to month to years. I refer to these visits as laundry “pegs” because the appearance of the charts resembles drying laundry on a clothesline.
I often get out of long-term trades when the clothesline is hit. Or at least I scale out some. Stocks do actually rally past their clotheslines sometimes but more often than not this represents a final climax moves before a long term top.
Please read some more about the clothesline here:
https://twitter.com/i/moment_maker/preview/928260044256407553
or here in my first article: https://thweis.com/how-to-set-up-effective-stock-market-graphs/#Logarithmic-scaling
The NDX follows two such trend lines PERFECTLY and anyones how ignores the concept doesn’t understand the sound foundation explained above.

Do you adjust the clothesline as the stock progresses? If so, how? What are the rules, please. Thank you
I would only do this if another price peak exceeds the second one. However I can’t remember doing this in real trading. The clothesline is normally set once you have to major peaks. Sometimes you can go to a lower timeframe such as the 1 hour chart to check if there is a level which fits better but those adjustments are small.