Were you asking for a precise value for the Tan within Tan problem? If
so, the answer is

1+ 1/5 Sqrt[11 + 6 Sqrt[2] + 8 Sqrt[1 + Sqrt[2]] -4 Sqrt[2
(1+Sqrt[2])]] =1.9617900032

The geometry is straightforward and I solved the resulting equation
with Mathematica.
I suspect you want something more than this, but as it was the only
problem on your site that I could readily solve, I dove in.

Thanks for a really wonderful website which has provided me hours of fun.

Best,

Bruce Schechter


I thought this 3-tans configuration is not so difficult to calculate in 
this configuration.
It seems obvious to me that in the bottom left corner the 90-deg. vertex 
of the large tan must coincide with the 45-deg. vertex of the smaller one.
Suppose that the 90-deg. vertex of the small triangle is at coordinates
  {Cos[alpha],Sin[alpha]}
then it is not so difficult to calculate the horizontal base of the 
larger tan as
  Sqrt[2]+2Cos[alpha]
and the vertical one as
  1+Sin[alpha]
These are equal for
  alpha = Arccos[(2-Sqrt[8]+Sqrt[2+Sqrt[8]])/5] = 1.29347..
Then the sides of the larger tan equal
  (4+Sqrt[2]+Sqrt[8*(1+Sqrt[2])])/5 = 1.96179..
which is a bit less than the 1.966 that is mentioned on the site.

Unless you meant that there is a completely different configuration that 
does the trick...

Greetings,
-- 
   Dave Langers.