Hans is as old as Peter will be (x years from now)

H = P + x

(x years from now is) when Hans will be twice as old as Peter was (y years ago)

H + x = 2(P - y)

(y years ago was) when Hans was half as old as the sum of their current ages.

H - y = (H + P)

Peter is as old as Hans was (z years ago) 

P = H - z

(z years ago was) when Peter was half as old as he will become over ten years.


P - z = (P + 10)

We simplify the equations:

H = P + x                 or H - P - x = 0
H + x = 2(P - y)          or H + x = 2P - 2y  or H - 2P + x + 2y = 0
H - y = (H + P)         or 2H - 2y = H + P  or H - P - 2y = 0
P = H - z                 or -H + P + z = 0
P - z = (P + 10)        or 2P - 2z = P + 10 or P - 2z = 10

When simplified the five equations become

 H -  P - x           =  0
 H - 2P + x + 2y      =  0
 H -  P     - 2y      =  0
-H +  P          +  z =  0 
      P          - 2z = 10

That's 5 equations in 5 unknowns.  You can solve it by substitution
or by elimination or by matrix methods.  The solution is

H = 40, P = 30, x = 10, y = 5, z = 10

If you need help solving that 5×5 system of equations, post
again asking how.

Edwin