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.