Road bike wheel bulding and Law of cosines

I'm preparing for buildings wheels for my road bike.
This week I ordered rims from Bike24, a bike shop in Germany.
If I'm lucky enough, I'm going to get the parts I ordered within a week.
Then I'm going to order hubs, measure the dimensions of rims and hubs, calculate spoke lengths, and then order the spokes.
It won't be until those spokes arrive that I really begin to lace them.
There are given formulas to calculate lengths and even online calculators in which the lengths are automatically calculated after you choose rims and hubs, if their dimension data have already been registered.
However, just out of curiosity, I wanted to calculate formula by myself, drawing figures. This is a simple problem of plain geometry, but at first I struggled with it for a week, unable to find a clue. Then I recalled "The Law of cosines". I learned this law when I was in the first year of high school, in a math class. At that time it wasn't so difficult to understand, but more than 20 years later, I have completely forgotten how to lead this law.
After try and error, I figured out again, at first making some wrong or redundant calculations.

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