There is a survey [article][1] by Berndt, Choi, and Kang devoted to the set of 58 Ramanujan's problems. Many of the problems have generated serious research activity since they were republished  in *Ramanujan's Collected Papers* in 1927.

An elementary solution to the specific geometric problem you've mentioned can be found in  [*Ramanujan's Notebooks, Part III*][2] by Berndt (Springer, 1991, pp. 244-246). The problem stems from Ramanujan's work on modular equations of degree 3... 


  [1]: http://www.google.com/books?hl=en&lr=&id=TT1T8A94xNcC&oi=fnd&pg=PA215&dq=related:DJmDgKBpMq0J:scholar.google.com/&ots=ujqGzfahmA&sig=FpslhlMUpn6dFwItc5FINa7XYE0#v=onepage&q&f=false
  [2]: http://books.google.com/books?id=wLJ-Dtj193MC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false