The derivation and theorem are correct. Here are some short notes on the theorem:

Theorem: Let a, b, and c be three numbers with a sum of X. Let W be the average of the first two numbers (a and b), and let P be the average of the last two numbers (b and c). Then:

• The first number a is equal to (2W - P - X) / 2.
• The second number b is equal to (2W + 2P - X) / 2.
• The third number c is equal to (2P - W - X) / 2.
• The sum of the three numbers (a + b + c) is equal to X.

These formulas can be useful when working with problems involving averages and sums of multiple numbers. By knowing the average of certain subsets of the numbers, we can derive the value of each individual number. And by checking that the sum of the derived numbers is equal to the given sum X, we can verify that our calculations are correct.

It's worth noting that the formulas can also be rearranged to find W and P, given a, b, and c. For example:

• W = (a + b) / 2
• P = (b + c) / 2

Overall, the theorem provides a useful tool for solving problems involving sets of numbers with known averages and sums.