Solution

Denote the squares with letters A to I.

We have . To have an equal sum of each row, we divide 45 by 3 and obtain 15. So the sum of each row, column and diagonal is 15.

Then observe   

We then consider the parity of the numbers at corners.

Consider the case if A is odd.

(1)    If B is also odd:

Then since , C is odd as well. As  and , G is also odd. By similar idea, D, I and H are also odd. But now we have 6 odd numbers, which is impossible. So B is not odd if A is odd.

(2)    If B is even:

Since , C is even. Similarly, H is even. Then  means that I is odd. This further gives F and G are even. Now we have 5 even numbers, which is impossible. So B is not odd if A is odd.

Since the above two cases are impossible, A must not be odd. It is even.

Similarly, C, G and I cannot be odd. They are even.

Fill in the 4 even numbers, 2, 4, 6 and 8, in the 4 corners and find the corresponding the values of B, D, F and H.

Here shows one of the solutions: