Where are the angles?!

Consider my approach...

Join P, Q, Rand S...

Now join AC and DB...

Since, SR is a line joining the midpoints of two sides of a triangle, it will be parallel to the third side...

Moreover...

$\frac{DS}{SA}$ = $\frac{DR}{RC}$ = $\frac{SR}{AC}$ = $\frac{1}{2}$

Therefore, SR is half of AC...

The same can be done for SP, PQ and QR..

After this we will get that SR = PQ = $\frac{SR}{AC}$

and, SP = RQ = $\frac{RQ}{DB}$

Now we get that PQRS is a parallelogram but it's diagonals are equal which proves that PQRS is either a rectangle or a square..

... but either of the two will give angle SPQ = $\boxed{90*}$

hence proved....