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> If $a, b, c$ are in geometric progression and $\log a -\log 2b, \log 2b -\log 3c, \log 3c-\log a$ are in AP. Find the type of the triangle, if it’s sides are $a, b, c$
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From the given data

$$b^2=ac$$ and 
$$2(\log 2b-\log 3c)=\log 3c-\log a +\log a -\log 2b$$

$$\log 2b=\log 3c$$
$$2b=3c$$

How does this tell us the type of the triangle? The relations between the sides is $$2b=3c$$and $$4a=9c$$

$$2b=3c=\frac{4a}{3}$$