3x3 Magic Square of 8 Squares
Construction Formula

The Derivation
Here is the well-known 3-term formulation.
   c+b      c-(a+b)  c+a               A2  B2  C2
   c-(b-a)  c        c+(b-a)    ===>   D2  E2  F2
   c-a      c+(a+b)  c-b               G2  H2  I2

There are eight 3-square arithmetic progressions, four of which go through
the center.  This construction deals with the other four.
   B2 + a = I2;  I2 + a = D2
   B2 + b = G2;  G2 + b = F2
   D2 + b = C2;  C2 + b = H2
   F2 + a = A2;  A2 + a = H2
which is equivalent to
   (1)  B2 + D2 = 2I2
   (2)  B2 + F2 = 2G2
   (3)  H2 + D2 = 2C2
   (4)  H2 + F2 = 2A2
   (5)  D2 - B2 = H2 - F2

Equation (5) forces (1) and (4) to have the same step value.
It also makes the step values of (2) and (3) the same.


The Construction
Equations (1)-(4) can be satisfied using the following formula.
It does not give all solutions, just most of them, including the smallest one.

Given any two different 3-square arithmetic progressions
   p2 + q2 = 2r2
   t2 + u2 = 2v2
scale each twice by multiplying by each end-term of the other.
   (pt)2 + (qt)2 = 2(rt)2
   (pu)2 + (qu)2 = 2(ru)2
   (pt)2 + (pu)2 = 2(pv)2
   (qt)2 + (qu)2 = 2(qv)2

Rearrange the terms and equations so that
they match the format of equations (1)-(4).
   (pt)2 + (qt)2 = 2(rt)2
   (pt)2 + (pu)2 = 2(pv)2
   (qu)2 + (qt)2 = 2(qv)2
   (qu)2 + (pu)2 = 2(ru)2


Example
The two smallest 3-square arithmetic progressions are
   12 +  72 = 2x 52
   72 + 172 = 2x132
The formula yields
    72 + 492 = 2x352
    72 + 172 = 2x132
  1192 + 492 = 2x912
  1192 + 172 = 2x852

I won't bother putting these entries into the magic square
because equation (5) gives
   492 - 72 = 1192 - 172
which isn't true.


The Proof
In order for the construction formula to satisfy equation (5), we must have
   (qt)2 - (pt)2 = (qu)2 - (pu)2
or
   (t2)(q2 - p2) = (u2)(q2 - p2)
or
   (t2 - u2)(q2 - p2) = 0.

So either t = u or q = p.
But if t = u, then the two entries (pt) and (pu) are duplicates.
And if q = p, then the two entries (pt) and (qt) are duplicates.
Therefore, the construction formula can never make a magic square of
distinct entries.


Conclusion
This is an important result, both for future computer searching and
for impossibility proofs since it makes a huge reduction in the problem.
This is because most of the solutions to equations (1)-(4) have this
construction. They have now been eliminated.