Construct a segment with length of $\frac{2a}{a+b^2}$

Question

Given the following segments how would you construct a segment with length of $\frac{2a}{a+b^2}$? Given the three line segments below, of lengths a, b and 1, respectively:

For example if I wanted to construct a segment $z$ satisfying an equation $a^2=b^2-z^2$ I could create a right triangle but I'm kind of stuck on what geometric figure to try to play with given this length.

Answer 1

Given a length defined as $1,$ we can construct any length that is described by addition, subtraction, multiplication, division, and taking square root of positive elements.

Answer 2

Without giving the full answer away, the following diagram should give you a hint of one way to use the steps given by Will Jagy to obtain the required segment. You will need to figure out the lengths of the segments indicated by the grey boxes.