Solve-By-Play

Well, how we solve the puzzle from the previous post?

$ maple
License expires in 22 days
> solve([3*a*b-c*d=0,2*(a+b)-3*2*(c+d)=0, a>0, d>0]);
     d (a - 3 d)      3 a (a - 3 d)
{b = -----------, c = -------------, 0 < a, 0 < d}
       9 a - d           9 a - d

Let's verify it.

$ ginsh
> a=1; d=1;
1
1
> b=(d*(a-3*d))/(9*a-d);
-1/4
> c=(3*a*(a-3*d))/(9*a-d);
-3/4

Ooops. But wait!
Plot of the rectangle puzzle

> a=1; d=10;
1
10
> b=(d*(a-3*d))/(9*a-d);
290
> c=(3*a*(a-3*d))/(9*a-d);
87
> (c*d)/(a*b);
3
> (2*(a+b))/(2*(c+d));
3

So, (1, 290) and (10, 870) is one possible solution. But there are
infinitly many!

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