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The geometric mean proportion between two numbers 'a' and 'b' is given by $$\sqrt{a\times b}$$
The given two numbers are 30 + $$\sqrt200$$ and 54 - $$\sqrt648$$
which are also equal to 10(3+$$\sqrt2$$) and 18(3-$$\sqrt2$$).
Hence the geometric mean proportion = $$\sqrt {10(3+\sqrt2)\times 18(3-\sqrt2)}$$ = $$\sqrt {180\times (3^2 - (\sqrt2)^2)}$$ = $$\sqrt {1260}$$ = $$6\times \sqrt35$$.
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