Magic Square Diagonals

The central number always has to be $\tfrac 13$ of the magic sum $S$ . If the number in one corner is $a$ then the opposite corner must have the number $b = S - \tfrac 13 S - a = \tfrac 23 S - a \Leftrightarrow a+b=\tfrac 23S \Leftrightarrow \tfrac {a+b}2 = \tfrac 13 S$ .

In an arithmetic progression, every number is the arithmetic mean of its two neighbours, this is satisfied here, so the three numbers form an arithmetic progression.