jmc

Let the two numbers be $a$ and $b$. If they have arithmetic mean $2700,$ then $$\frac{a+b}{2}=2700,$$which gives us $a+b=5400$. Since their harmonic mean is $75$ we have $$\frac{2}{\frac{1}{a}+\frac{1}{b}}=75.$$We can rearrange terms to get $$\frac{1}{a}+\frac{1}{b}=\frac{2}{75}.$$Taking the common denominator gives us $$\frac{a+b}{ab}=\frac{2}{75}.$$Substituting the value of $a+b$ and solving for $ab$ gives $$ab=\frac{5400\cdot 75}{2}=2700\cdot 75.$$Then the geometric mean is $$\sqrt{ab}=\sqrt{2700\cdot 75}=\boxed{450}.$$