Not an answer, just an illustration of the uniqueness as mentioned in Fedor Petrov's comment:
[![Pent.gif][1]][1]
Edge lengths: $1, 3, 4, 2, 6$. Final circle radius $\approx 3.045$.
[![Pent.gif][1]][1]
Edge lengths: $1, 3, 4, 2, 6$. Final circle radius $\approx 3.045$.
Joseph O'Rourke
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