What are the semiaxes a, b, c, of an ellipsoid

Question

with girths $G_{bc}, G_{ca}, G_{ab}$? E.g., if we knew the semiaxes were $$a=1, b=1, c=1/√2,$$ the three axis-aligned cross-sections would be ellipses with semiaxes

In[54]:= Permutations[{1, 1, 1/√2, {2}]

Out[54]= {{1, 1}, {1, 1/√2}, {1/√2, 1}}

and girths

In[55]:= ArcLength[Circle[{, }, #]] & /@ %

Out[55]= {2 π, 2 √2 EllipticE[-1], 4 EllipticE[1/2]}

In[56]:= FunctionExpand@%

Out[56]= {2 π, (4 π^(3/2))/Gamma[1/4]^2 + (8 Gamma[5/4]^2)/ √π, π^(3/2)/Gamma[3/4]^2 + (2 Gamma[3/4]^2)/√π}

But how can we recover a, b, and c from these three girths?