Constructing $ab$ from $a$ and $b$ alone. I would like to know if we are given two arbitrary line segments having lengths, $a$ and $b$, is it possible to construct a line segment having length a\cdot b without defining a unit length?
To construct a line having length equal to the mean proportional of $a$ and $b$, $\sqrt {ab}$, does not require a standard unit length to be defined.  So, it makes me wonder if the same can be done with $ab?$
And related to this construction, is it possible to construct the length equal to $a^2$ given a line segment of length $a?$
If so, could you please say yes/no, then some space afterwards with a hint, and finally more space for the answer.  I would like to find the construction for myself.
And if not, could you supply a proof as to why?  I can reason as to why.  If length $a$ is less than $1$, then $a\cdot a$ will be even less so: $a^2 < a$. But the only way to know this is to know the unit length.  Hence, you will need to define a unit length ahead of time.  
Or so my reasoning goes, but I would think the same reason would also apply to the mean proportional, yet for the mean, no unit length is needed, why should these two constructions differ? They seem so very close.
Thanks.  
 A: If you don't have a unit length, then a necessary condition to be able to construct a function $f(a,b)$ of any two lengths $a$ and $b$ is scale-invariance:$$f(\lambda a, \lambda b) = \lambda f(a,b).$$  
In your examples, $\sqrt{ab}$ is scale-invariant, but $ab$ and $a^2$ are not.  The reason is because if $f$ is not scale-invariant, then if you change the unit, the correct segment $f(a,b)$ will be different.
A: No. Suppose you had such a construction. If I shrink your picture by a factor of $2$, then the construction will still work out. However, this is bad because it means that the starting lengths scale to $a/2$ and $b/2$, but the ending length scales to $ab/2$. However, $ab/2\neq a/2\cdot b/2$, so this doesn't work.
Essentially, in order for a construction to exist, everything must scale together - however, a product scales with the square. You can construct the length $\frac{ab}c$ from the lengths $a,\,b$ and $c$, since this scales linearly as the three given lengths scale together. Basically, you usually take $c=1$ to get your ordinary operations - but you must recognize that this means that you can no longer scale your diagrams.
A: Suppose a=1 and b=1. Then you can make ab=1 work. But 1 what? Metre? Unit? Without a unit length there is no way to tell.
Suppose that you measured in centimetres instead of metres. Then a=100 and b=100, so c would need to be 10000 or 100 metres. But a and b are exactly the same as before! This is bad, as c=1 metre or c=100 metres. In fact we could make c any distance by choosing an appropriate 'unit' length.
A: I think part of the issue is that the there are actually two different definitions and concepts of "multiplication" and "square root": via linear values, and via area values.
The value $ab$ is the area value of a rectangle that is has two sides $a$ (to some scale) and $b$ (to the same).  This can be thought of as a binary function: $A: (\mathbb R, \mathbb R) \to \mathbb R^2$.  
But the value $a*b$ as a linear value is the length that a side of a rectangle must be if the area of a $1 \times a*b$ rectangle will have the same area as a $a \times b$ rectangle will have.  This can be though of as a binary function: $M:(\mathbb R, \mathbb R) \to \mathbb R$.
The relationship between $A$ and $M$ is $A(a,b) = A(1, M(a,b))$.
Likewise there are two different concepts of square root.
$\sqrt{ab}$ is the square root of an area value.  i.e. $R:\mathbb R^2 \to \mathbb R$.  It by definition, the length that the side of a square must be so that it will have the same area as a $a\times b$ rectangle. 
$\sqrt{a*b}$ would be the square root of a linear value. $S:\mathbb R \to \mathbb R$.  That is $\sqrt{k}$ would be, by definition, the length of the side of the square must be so that it will have the same area as a $1\times k$ rectangle.
I think a huge part of the confusion is that our construction is for $\sqrt{ab}$ and does not require a unit value, but it looks like $\sqrt{a*b}$ and as $a*b$ requires are unit value, how could $\sqrt{a*b}$ not require it?
Well, because our construction was an entirely different operation.
